RICHARD SCHOEN - SPECTRAL GEOMETRY (MATH 286 ...

RICHARD SCHOEN - SPECTRAL GEOMETRY (MATH 286) 

LECTURE NOTES 

NOTES BY OTIS CHODOSH 

Contents 

1. Classical Eigenvalue Theory 2 

1.1. Eigenvalues and Isoperimetric Problems 3 

1.2. Questions about Eigenvalues 5 

1.3. Upper Bounds on λk 6 

1.4. Lower Bounds for λk 8 

2. Sharp Lower Bounds on λ1 10 

2.1. The Fundamental Gap Problem 14 

3. Background on Surfaces and Harmonic Maps 14 

3.1. Area and Energy of Maps 15 

3.2. Harmonic Maps 16 

4. Conformal Balancing of Measures 18 

4.1. First Eigenvalue Bounds by Conformal Balancing 19 

4.2. Coarse Upper Bounds By Conformal Balancing 23 

4.3. Conformal Volume 24 

5. Free Boundary Minimal Surfaces and Steklov Eigenvalues 28 

5.1. Examples of Steklov-Critical Submanifolds 29 

5.2. Structure of Extremal Metrics 30 

References 30 

Date: February 7, 2013. 

1

2 NOTES BY OTIS CHODOSH 

For reference, the class website is http://math.stanford.edu/~schoen/math286. These notes 

are very much a work in progress on my part, caveat emptor. 

Please inform me at ochodosh@math.stanford.edu of errors of any magnitude. 

1. Classical Eigenvalue Theory 

The main reference for this section is [SY94, Chapters 3 and 4]. We’ll first consider Riemannian 

manifolds (M n , g). One problem we’ll discuss is maximizing the first eigenvalue of the Laplacian, 

λ1(∆g) subject to the area constraint A(g) = 1. Recall that 

 

(1.1) λ1(g) = min |∇u| 2 

dVg : u 2 

= 1, 

This is the first eigenvalue of the Laplacian associated to g 

(1.2) ∆gu = 

M 

1 

√ det g 

∂ 

∂x i 

M 

M 

det gg ij ∂u 

∂x j 

 

 

u = 0 . 

There are several boundary conditions we could associated with this problem, depending on whether 

or not ∂M is empty. If ∂M = ∅, then we have an increasing sequence of eigenvalues 0 = λ0 < λ1 ≤ 

λ2 ≤ . . . (written with multiplicity). If ∂M = ∅, then we have several natural choices of boundary 

conditions 

Definition 1.1. (Dirichlet Boundary Conditions) Here, we fix u = 0 on ∂M. This results in 

eigenvalues 

0 < λ D 1 < λ D 2 ≤ λ D 3 . . . 

Definition 1.2 (Neumann Boundary Conditions). We consider admissible functions with ∂u 

∂ν = 0 

on ∂M, where ν is a normal vector along the boundary. This gives eigenvalues 

0 = λ N 0 < λ N 1 ≤ λ N 2 ≤ . . . 

We will later consider the following eigenvalue problem as well 

Definition 1.3 (Steklov Eigenvalues). These are the spectrum of the Dirichlet to Neumann map 

where Lu = ∂û 

∂ν 

L : C ∞ (∂M) → C ∞ (∂M), 

(where ν is the outward pointing normal to ∂M) for û solving the Dirichlet problem 

 

∆û = 0 in M 

û = u on ∂M 

It is not hard to see that the eigenvalues take the form σ0 = 0 < σ1 ≤ σ2 ≤ . . . . 

As briefly mentioned above, we have the following variational characterization of the eigenvalues: 

(1) If ∂M = ∅ or we are using Neumann boundary conditions, then 

 

|∇u| 2 

λk = min 

M u2 

 

: u ⊥ u0, . . . , uk−1 

(2) For Dirichlet boundary conditions, this is also valid, as long as we only consider u satisfying 

u = 0 on ∂M. 

(3) The Steklov eigenvalues have a similar characterization 

 

|∇u| 2 

 

σk = min : uui = 0, i = 1, 2, . . . , k − 1 . 

u2 

∂M

RICHARD SCHOEN - SPECTRAL GEOMETRY (MATH 286) LECTURE NOTES 3 

1.1. Eigenvalues and Isoperimetric Problems. There are several isoperimetric constants which 

we now define 

Definition 1.4 (Isoperimetric Constants). We let 

 

 

c1 = inf 

|∂Ω| 

: Ω ⋐ M, Ω has finite perimeter 

|Ω| n−1 

n 

denote the usual isoperimetric constant. Additionally we define (if ∂M = ∅) 

 

|∂Ω| 

hD(M) = inf : Ω ⋐ M 

|Ω| 

 

 

|∂Ω| 

hN(M) = inf 

: ∂Ω is a hypersurface separating M into two pieces 

min{|Ω|, |M\Ω|} 

It is not hard to check that if ∂M = ∅, then 

 

(1.3) c1 

M 

|u| n 

n−1 

n 

n−1 

 

≤ 

M 

|∇u|. 

Furthermore if ∂M = ∅, then 

 

hD(M) |u| ≤ |∇u| for all u with u = 0 on ∂M 

M M 

 

hN(M) |u| ≤ |∇u| 

M M 

for all u ≥ 0 with |{u = 0}| ≥ 1 

2 |M|. 

(1.4) 

These inequalities are ready consequences of the co-area formula 

Proposition 1.5 (Co-Area Formula). For M a compact Riemannian manifold, possibly with boundary, 

if f ∈ H1 (M), then for any measurable function g on M we have 

 

∞ 

g 

g = 

dσ. 

|∇f| 

M 

−∞ 

{f=σ} 

Because it is more difficult, we check the second inequality in (1.4). For u ≥ 0, note that 

|{u = 0}| ≥ 1 

2 |M| implies that |{u ≥ a}| ≤ |{u ≤ a}|. Then, by the co-area formula 

 

∞ 

|∇u| = 

1 da 

M 

0 

∞ 

{u=a} 

(1.5) 

= 

0 

|∂{u ≥ a}|da 

∞ 

≥ hN(M) |{u ≥ a}|da 

0 

= hN(M) u. 

The above isoperimetric inequalities are closely related to the first eigenvalue, as first observed 

by Cheeger 

Theorem 1.6 (Cheeger). For M compact with nonempty boundary, we have 

λ D 1 (M) ≥ 1 2 

hD(M) 

4 

and 

λ N 1 (M) ≥ 1 

4 hN(M) 2 . 

M


Proof. For the first inequality, we’ll show that for all u with u|∂M = 0 then 

 

1 2 

(1.6) 

hD(M) u 

4 2 

≤ |∇u| 2 . 

To do so, we replace u by u2 in the first equation in (1.4), above. This gives 

(1.7) 

 

hD(M) u 2 

≤ 

 

2|u||∇u| ≤ u 2 

1 

2 

M 

M 

M 

M 

M 

|∇u| 

M 

2 

1 

2 

, 

which implies the desired inequality upon rearranging terms. 

For the second inequality, we’ll show that for any Lipschitz function f on M 

1 

(1.8) 

4 hN(M) 

 

|f − f| 2 

≤ |∇f| 2 . 

M 

We clearly have that |{f < f}| ≤ 1 

1 

2 |M| and |{f > f}| ≤ 2 |M|. Thus, applying (1.4), we see that 

 

hN(M) ((f − f)+) 2 

≤ |∇((f − f)+) 2 | 

(1.9) 

Thus 

(1.10) 

M 

1 

4 hN(M) 

 

 

= 2 

{f≥f} 

{f≥f} 

M 

(f − f)+|∇f| 

 

≤ 2 ((f − f)+) 

M 

2 

((f − f)+) 2 

≤ 

{f≥f} 

1 2 

|∇f| 2 . 

|∇f| 

{f≥f} 

2 

We may similarly show 

1 

(1.11) 

4 hN(M) 

 

((f − f)−) 2 

≤ |∇f| 

{f≤f} 

2 . 

Adding the two equations, and using the fact that 

{f=f} |∇f| = 0, as is easily checked, we may 

thus conclude our desired inequality. 

Theorem 1.7 (Faber-Krahn). For Ω ⊂ R n a smooth bounded domain, we have that for R > 0 so 

that |BR(0)| = |Ω|, then 

λ D 1 (Ω) ≥ λ D 1 (BR(0)) 

with equality if and only if Ω is a round ball. 

Proof. Let f be the first eigenfunction, with f > 0 in Ω and 

1 

2 

Ω f 2 = 1. We’ll define a symmetric 

rearrangement as follows. Choose a function on BR(0), g(x) which is a function of the radial 

variable only, i.e. g(x) = g(|x|), and where we require g = 0 on ∂BR and |{g ≥ c}| = |{f ≥ c}| for 

c ≥ 0 (it is clear that we can choose such a g). We claim that 

(1) 

Ω f 2 = 

BR g2 

(2) 

BR |∇g|2 ≤ 

Ω |∇f|2 

Taking these for granted for the moment, we see that this finishes the proof, becase 

(1.12) λ D 

BR 

1 (BR) ≤ 

|∇g|2 

 

 

|∇f| 2 = λ D 1 (Ω). 

≤ 

g2 

BR 

Now, we prove these two properties. For the first property, we see that 

 

(1.13) 

f 2 ∞ 

= |{f 2 ∞ 

≥ c}|dc = |{g 2 

≥ c}|dc = g 2 

Ω 

0 

0 

Ω 

BR


To see the second property, note that because g is radial, it is easy to check that |∇g| is constant 

on {g = c}. Thus 

(1.14) |{g = c}| = 

On the other hand, 

(1.15) 

 

{f=c} 

|∇g| 

{g=c} 

1 

2 

{g=c} 

 

1 

|∇f| 

{f=c} |∇f| ≥ 

2 1 

{f=c} 

= |{f = c}| 2 

≥ |{g = c}| 2 

 

= |∇g| 

{g=c} 

1 

2 

1 

|∇g| 

{g=c} 

1 

|∇g| . 

In the second to last line, we used that the surfaces {f = c} and {g = c} enclose the same area, 

and {g = c} is a sphere, so the isoperimetric inequality gives the desired bound. However, by the 

co-area formula 

(1.16) 

 

 

1 d 

d 

1 

= − |{f ≥ c}| = − |{g ≥ c}| = 

{f=c} |∇f| dc dc {g=c} |∇g| , 

so combining the last two steps, we see that 

 

 

(1.17) 

|∇f| ≥ |∇g|. 

Ω 

0 

{f=c} 

{g=c} 

We may finally conclude the proof by using the co-area formula 

 

(1.18) 

|∇f| 2 

∞ 

∞ 

= 

|∇f| dc ≥ 

|∇g| dc = 

{f=c} 

0 {g=c} 

BR 

|∇g| 2 . 

We remark that it is easy to see from the proof that equality holds if and only if Ω is a round 

ball. 

1.2. Questions about Eigenvalues. 

Question 1.8. We thus are led to three natural questions 

(1) What sort of bounds can we give on the first (low) eigenvalues? 

(2) What sort of bounds can we give for high eigenvaluse? 

(3) What geometric invariants are relevant? 

We remark that for the first eigenvalue, if Ric(g) ≥ (n − 1)k and diam(M) ≤ d then we can 

bound λ1 above and below in terms of k and d. Furthermore, the Weyl asymptotics tell us that for 

c(n) = (2n) −n ωn (where ωn = |B1|), we have that 

(1.19) λk(M) n 

2 ∼ c(n) k 

|M| . 

We further note 

Conjecture 1.9 (Polya). It is a classical conjecture by Polya that for Ω ⊂ R 2 

and 

λ D k 

λ N k 

(Ω) ≥ c(2) k 

|Ω| 

(Ω) ≤ c(2) k 

|Ω| .


Remark 1.10. It is known that for Ω ⊂ R n , 

1.3. Upper Bounds on λk. 

λk(Ω) n 

2 ≥ n k 

c(n) 

n + 2 |Ω| . 

Theorem 1.11. For (M, g) a complete manifold with Ric ≥ (n − 1)k, for x0 ∈ M letting B(x0, r0) 

denote the geodesic ball of radius r0 and Vn(k, r0) denote a ball of radius r0 in M n k , the simply 

connected manifold with constant sectional curvature k, then we have that 

λ D 1 (B(x0, r0)) ≤ λ D 1 (Vn(k, r0)) 

with equality if and only if B(x0, r0) isometric to Vn(k, r0). 

For ϕ(r) the first Dirichlet eigenvalue on Vn(k, r0) (it is radial by symmetry), notice that ϕ(r) > 0 

for r < r0 and ϕ ′ (r) ≤ 0 for r ≤ r0. Further note that the eigenvalue equation simplifies because of 

the symmetry of ϕ, giving 

(1.20) 

1 

′ 

′ det g(r)ϕ (r) + λ1ϕ = 0. 

det g(r) 

The second term is positive, and thus we have that 

1 

′ 

′ 

(1.21) 

det g(r)ϕ (r) < 0 

det g(r) 

for r < r0. 

Proof of Theorem 1.11. We let p(x) = dist(x, x0). Consider ϕ ◦ p ∈ Lip(B(x0, r0)) with ϕ ◦ p = 0 

on ∂B(x0, r0). We claim that 

 

(1.22) 

|∇(ϕ ◦ p)| 

B(x0,r0) 

2 ≤ λ D 

1 (Vn(k, r0)) (ϕ ◦ p) 

B(x0,r0) 

2 . 

To show this, we will pull the problem back to the tangent bundle by the exponential map (using 

expx0 (Br0 (0)) = B(x0, r0)). For each ζ ∈ Tx0M with |ζ| = 1, let a(ζ) ∈ (0, r0] be the maximum 

value so that expx0 (tζ) minimizes up to a(ζ). It is a standard fact that a(ζ) is continuous. We now 

define Ω = {rζ ∈ Tx0M : r ≤ a(ζ)}. Furthermore, notice that in Ω, p(ξ) = |ξ| is smooth. Now, we 

have that 

 

|∇(ϕ ◦ p)| 2 a(ζ) 

= 

|∇(ϕ ◦ ρ)| 2 (t, ζ)t n−1 

θ(t, ρ)dt dσ(ζ) 

(1.23) 

B(x0,r0) 

 

B(x0,r0) 

(ϕ ◦ p) 2 

= 

S n−1 

S n−1 

0 

a(ζ) 

(ϕ ◦ ρ) 2 (t, ζ)t n−1 

θ(t, ρ)dt dσ(ζ) 

0 

where θ(t, ζ) is the coefficient of the volume element in these coordinates. By the Volume Comparison 

Theorem (cf. [SY94, Theorem 1.2]) we have 

−1 ∂θ 

(1.24) θ 

∂t 

Thus 

(1.25) 

a(ζ) 

0 

2 dϕ 

t 

dt 

n−1 θ(t, ζ)dt = − 

a(ζ) 

0 

∂θk 

≤ θ−1 

k ∂t . 

 

n−1 −1 d 

t θ(t, ζ) ϕ t 

dt 

n−1 θ(t, ζ) dϕ 

 

t 

dt 

 

n−1 θ(t, ζ)dt 

:=I 

 

 

t=a(ζ) 

. 

t=0 

+ ϕ dϕ 

dt tn−1θ(t, ζ) 

 

=0


We further compute 

(1.26) 

 

d2ϕ n − 1 

I = ϕ + 

dt2 t 

 

d2ϕ n − 1 

≥ ϕ + 

dt2 t 

 

dθ dϕ 

+ θ−1 

dt dt 

 

dϕ 

+ θ−1 

k 

dθk 

dt 

dt 

 

=∆ M n k ϕ 

= −λ D 1 (Vn(k, r0))ϕ 2 

Inserting this into the above expression gives the desired inequality. Furthermore, if there is equality, 

we see that Ω = B(x0, r0) and Jacobi field arguments show that B(x0, r0) has constant curvature. 

 

We now recall the following general principle 

Proposition 1.12. For M compact without boundary and Ω1, . . . , Ωk+1 disjoint domains in M, 

we have that 

λk(M) ≤ max λ D 1 (Ω1), . . . , λ D 1 (Ωk+1) . 

Proof. For ϕ1, . . . , ϕk+1 the first eigenfunctions of Ω1, . . . , Ωk+1, we have that they all have disjoint 

support. Now, if {1, u1, . . . , uk−1} are the first k − 1 eigenfunctions for M, then (because it is an 

linear algebra problem) we may choose a1, . . . , ak+1 ∈ R (not all zero) so that 

k+1 

(1.27) ϕ := 

i=1 

aiϕi 

is orthogonal to the span of {1, u1, . . . , uk−1}. Then, we have that 

(1.28) 

 

M 

|∇ϕ| 2 k+1 

= 

i=1 

k+1 

= 

i=1 

a 2 i 

 

|∇ϕi| 

M 

2 

a 2 i λ D 1 (Ωi) 

 

ϕ 

M 

2 i 

≤ max 

i {λ1(Ωi)} 

k+1 

= max 

i {λ1(Ωi)} 

i=1 

 

M 

a 2 i 

ϕ 2 , 

 

ϕ 

M 

2 i 

which is the desired conclusion. 

Using the previous two results, we thus obtain 

Theorem 1.13. For M compact and Ric ≥ (n − 1)k, then λm(M) ≤ λ1 

Vn 

 

diam M k, 2m . 

d 

Proof. If d = diam(M), then we can find m + 1 disjoint balls of radius 2m . Thus, we may apply 

the above two results. 

Corollary 1.14. For M compact and Ric ≥ (n − 1)k 

(1) If k = 0, then 

λm(M) ≤ n2 m 2 π 2 

d 2


(2) If k = 1 then 

(3) If k = −1 then 

λ1 ≤ 

λ1(M) ≤ nπ2 

d 2 

(n − 1)2 

4 

+ 2n(n + 4) 

d 2 

1.4. Lower Bounds for λk. The following lower bound for the first eigenvalue is classical 

Theorem 1.15 (Lichnerowicz). For (M, g) closed, Ric ≥ (n − 1)k (for k > 0) then λ1(M) ≥ nk. 

Although we will not prove it, we remark that Obata has proven that equality in the above 

theorem implies that M is isometric to a round sphere of sectional curvature k. 

Proof. Let u be a first eigenfunction normalized so that 

M u2 = 1. Then, we have the following 

“Ricci formula” at p ∈ M (letting ei be an orthonormal frame with [ei, ej]|p = 0 at p) 

(1.29) 

1 

2 ∆|∇u|2 = 1 

∇ei 

∇ei (∇eju)2 2 

= ∇ei (∇eju∇ei ∇eju) = |∇ 2 u| 2 + (∇eju)∇ei ∇ei∇ej u 

= |∇ 2 u| 2 + (∇eju)∇ei ∇ej∇ei u 

= |∇ 2 u| 2 + 〈∇u, ∇∆u〉 + Ric(∇u, ∇u). 

We note that until this step, we have not used anything about u or the metric. However, we may 

now use our lower Ricci bound, combined with the fact that u is an eigenfunction to see that 

1 

2 ∆|∇u|2 

 

≥ 

∇2u − ∆u 

n g 

 

2 

 

+ 1 

n (∆u)2 − λ1(M)|∇u| 2 + (n − 1)k|∇u| 2 (1.30) 

. 

Integrating this inequality and throwing away the first term on the right 1 , thus gives (using the 

fact that 

M u2 = 1) 

(1.31) 0 ≥ 1 

n λ1(M) 2 − λ1(M) 2 + (n − 1)kλ1(M). 

Rearranging this gives the desired inequality. 

Now we give a lower bound for all of the eigenvalues, which is not sharp but does have the right 

asymptotic behavior as compared to the Weyl asymptotics. We’ll do so for M compact of dimension 

≥ 3 with ∂M = ∅ and under the Dirichlet boundary conditions, although these restrictions can be 

removed by slightly modifying the technique used below. We recall that the isoperimetric inequality 

(1.32) c1|Ω| n 

n−1 ≤ |∂Ω| 

is equivalent to the L1-Sobolev inequality 

 

(1.33) c1 |ϕ| n 

n−1 

M 

n−1 

n 

 

≤ 

We will instead consider the L2-Sobolev inequality 

(1.34) 

 

c2 |ϕ| 2n 

n−2 

n 

n−2 

 

≤ 

1 It is the vanishing of this term that Obata analyzes 

M 

|∇ϕ| for ϕ|∂M = 0. 

|∇ϕ| 2 

for ϕ|∂M = 0,


which we easily derive from the L 1 version by plugging in ϕ 2 . This gives the lower bound 

(1.35) c2 ≥ 

2 n − 2 c2 1 

n − 1 4 . 

We further remark that Croke [Cro80] has shown that if M n has Ric ≥ (n − 1)k, diam ≤ D and 

Vol(M) ≥ V , then we have that c1(M) ≥ C(n, d, k, V ). 

Theorem 1.16 ([CL81]). For M with nonempty boundary, and of dimension n ≥ 3, we have 

λ D k 

(M) ≥ c2 

e 

2 

k n 

. 

|M| 

Proof. The proof uses the heat kernel, whose essential properties we now recall. It is a function 

H(x, y, t) defined on M × M × (0, ∞) so which the fundamental solution to the heat equation, i.e. 

⎧ 

 

⎪⎨ 

∂ 

∂t − ∆x H = 0 

(1.36) 

H|x∈∂M = 0 

⎪⎩ 

limt↘0 H(x, y, t) = δy(x). 

In particular, it has the following properties 

(1) limt↘0 H(x, x, t) = ∞ 

(2) H(x, y, t) = H(y, x, t) (this follows from self-adjointness of the heat operator) 

(3) H(x, y, t)dy ≤ 1 (because of the Dirichlet boundary conditions) 

(4) If ui, λi are the eigenfunctions and eigenvalues of ∆, and we take the ui to be L 2 -orthonormal, 

then H(x, y, t) = ∞ 

i=1 e−λit ui(x)ui(y), where the sum converges (along with the term by 

term derivatives) for t > 0. 

(5) For 0 < s < t, H(x, y, t) = 

We note that the last property implies that 

 

(1.37) H(x, x, 2t) = 

M 

H(x, z, s)H(z, y, t − s)dz. 

M 

(H(x, y, t)) 2 dy. 

With this in mind, we compute 

 

∂ 

H(x, z, t) 

∂t M 

2 

= 2 H(x, z, t) 

M 

∂ 

(H(x, z, t)) dz 

∂t 

 

= 2 H(x, z, t)∆z (H(x, z, t)) dz 

M 

= −2 |∇zH(x, z, t)| 2 (1.38) 

dz 

≤ −2c2 

Now, using Hölder’s inequality, we have that 

(1.39) 

 

M 

H 2 

dz ≤ 

H 2n 

n+2 H 4 

 

n+2 ≤ 

M 

 

(H(x, z, t)) 

M 

2n 

n−2 

n 

n−2 

H 2n 

n−2 

n+2 

n−2 

 

Combined with the above inequality, this thus shows that 

(1.40) 

 

∂ 

∂t 

H 2 

dz ≤ −2c2 

H 2 n+2 

n 

dz 

4 

n+2 

H 

. 

 

≤ 

H 2n 

n−2 

n+2 

n−2 .


Letting f(x, t) = H(x, x, 2t), by (1.37), we thus see that 

(1.41) 

i=1 

∂f 

∂t 

n+1 

≤ −2c1f n 

limt↘0 f(x, t) = ∞, 

which may be integrated for a fixed x, showing that f(x, t) ≤ 4 

nc2t − n 

2 . Integrating this in x and 

using the fact that H(x, x, 2t) = ∞ i=1 e−2λit (ui(x)) 2 , we thus have that 

∞ 

(1.42) 

e −2λit 

 

4 

≤ 

n c2t 

n 

− 2 

|M|. 

Taking tk so that 2λktk = n 

2 

n 

− 

(1.43) ke 2 ≤ 

, we thus see that 

∞ 

e −2λitk 

 

4 

≤ 

n c2tk 

i=1 

− n 

2 

= 

c2 

λk 

− n 

2 

|M|, 

which yields the desired bound. 

2. Sharp Lower Bounds on λ1 

We’ll now discuss the recent results of Andrews-Clutterbuck, giving sharp lower bounds for the 

first eigenvalue on closed manifolds. The main references for this section are the works [AC11, 

AC12]. 

Theorem 2.1 (Andrews-Clutterbuck [AC11, AC12]). If we let 

λ1(n, k, D) = inf{λ1(M n , g) : M closed , Ric ≥ (n − 1)k, diam ≤ D}, 

then 

λ1(n, k, D) = µ(n, k, D) 

where µ is the first nonzero eigenvalue of the following 1-dimensional operator 

1 

Lϕ = 

(ck) n−1 

 

(ck) n−1 ϕ ′ ′ ′′ 

= ϕ − (n − 1)Tkϕ ′ 

on the interval [−D/2, D/2] with Neumann boundary conditions. Here 

⎧ 

⎪⎨ cos( 

ck(τ) = 

⎪⎩ 

√ kτ) k > 0 

1 k = 0 

cosh( √ −kτ) k < 0. 

We have also defined 

⎧√ 

√ 

⎪⎨ k tan( kτ) k > 0 

Tk(τ) = 1 

⎪⎩ 

− 

k = 0 

√ −k tanh( √ −kτ) k < 0, 

in order to have a more convenient expression for L. 

We briefly recall some of the history behind this estimate. For k = 0, this was proven by Zhong- 

Yang in 1985 [ZY84] (with the sharp constant), but the proof was much more difficult. For k > 0, 

we may bound diam ≤ π √ k by Meyer’s Theorem, and as we check below, this recovers Lichnerowicz’s 

theorem. However, for a smaller D this is sharper. For k = 0, some progress on this was made 

by Bakry-Qian in 2000 [BQ00]. We finally remark that recently Andrews-Ni [AN11] extended this 

work to the Bakry-Emry tensor and the associated f-Laplacian. 

Example 2.2. If k = 0, then the operator L becomes Lϕ = ϕ ′′ on [−D/2, D/2], so the first 

eigenfunction is ϕ(τ) = sin πτ 

D 

. Thus µ = π 2 

D 2 . As such λ1(n, 0, D) = π2 

D 2 .


Example 2.3. For k > 0, by Meyer’s Theorem, we have that D ≤ π 

√ D , as remarked above. Thus, 

the operator L becomes 

Lϕ = ϕ ′′ − (n − 1) √ k tan( √ kτ)ϕ 

so the first eigenfunction is ϕ(τ) = sin( √ kτ) (it is easily seen to be an eigenvalue, and Sturm- 

Liouville theory shows that it must be the first one). Thus, we easily see that λ n, k, π 

 

√ = nk, 

D 

which is simply Lichnerowicz’s result. 

The strategy of the proof is to detect the lowest eigenvalue by knowing how quickly solutions to 

the heat equation decay. This is because we may solve 

(2.1) 

∂u 

∂t 

= ∆u 

u(x, 0) = u0(x) 

by expanding u0 = ∞ 

i=0 aiϕi where ϕi are eigenvalues of the Laplacian. Then, the solution to the 

heat equation is u(x, t) = ∞ 

i=0 e−λit aiϕi. This does not necessarily converge to zero, as λ0 = 0, 

but the key idea is that |ϕ(x, t)−ϕ(y, t)| does converge to zero and in fact |ϕ(x, t)−ϕ(y, t)| ≈ e −λ1t . 

Thus, the main ingredient in Andrews-Clutterbuck’s proof is showing that 

|u(x, t) − u(y, t)| ≤ ce −µt . 

for any solution to the heat equation. Then, taking u = e −λ1t ϕ1(x), we see that 

|ϕ1(x) − ϕ1(y)| ≤ ce (λ1−µ)t , 

which implies that λ1 ≥ µ. 

The main ingredient in the proof of Theorem 2.1 is the following 

Theorem 2.4. Assume that (M n , g) is closed with Ric ≥ (n − 1)k and diam ≤ D. Fix a function 

ϕ(t) (defined for t ∈ [0, D/2]) with ϕ ′ ≥ 0. Let v(x, y) = 2ϕ 

d(x,y) 

2 

 

. Then, the linear operator, 

defined on functions M × M → R 

L (u) := inf trg(A Hessu) : A ∈ Sym 2 (T ∗ (M × M), A ≥ 0, A = g −1 on TxM and TyM 

satisfies 

L (v) ≤ 2Lϕ 

in the viscosity sense (where both sides are evaluated at d(x, y)/2). 

We will define what it means to satisfy the differential inequality “viscosity sense” later, as we 

will see that it follows immediately from the proof at smooth points of v. We further remark that 

(in coordinates) the assumption on A that we are making is: regarding A as a (symmetric) 2 × 2 

matrix of n × n blocks, the on-diagonal blocks are g −1 and the off diagonal blocks need only to be 

chosen so that A is symmetric and non-negative. In particular, trg(A Hessu) looks like the Laplacian 

on M × M but with additional cross terms. 

Proof. Assyme that (x, y) is a smooth point of v, i.e. that there is a unique minimizing geodesic from 

x to y, γ(s) for s ∈ [−d/2, d/2] (we will write d = d(x, y)). Pick an orthonormal basis {E (x) 

i }n i=1 for 

TxM so that E (x) 

n = γ ′ (−d/2). We may parallel transport the frame to y, obtaining {E (y) 

i }n i=1 

orthonormal basis for TyM so that E (y) 

n = γ ′ (d/2). These combine to give a basis for T (x,y)M × M. 

We fix 2 

n−1 

(2.2) A = (E (x) 

i , E (y) 

i ) ⊗ (E (x) 

i , E (y) 

i ) + (E (x) 

n , −E (y) 

n ) ⊗ (E (x) 

n , −E (y) 

n ). 

i=1 

2 We are identifying T ∗ M with T M via the metric. 

, an


It is easy to check that A is an allowable choice in the definition of L , and that it suffices to prove 

the desired inequality for A. We note that 

n−1 

(2.3) trg(A Hessv) = Hessv((E (x) 

i , E (y) 

i ), (E (x) 

i , E (y) 

i )) + Hessv((E (x) 

n , −E (y) 

n ), (E (x) 

n , −E (y) 

n )). 

i=1 

Furthermore, from the chain rule 

(2.4) Hessv = ϕ ′ Hess d(x,y) + 1 

2 ϕ′′ ∇d ⊗ ∇d. 

We thus use this to compute trg(A Hessv). We first let σ(r) = (γ(−d/2 + r), γ(d/2 − r)), which 

has σ ′ (0) = (E (x) 

n , −E (y) 

n ). But, d(σ(r)) = d(γ(−d/2 + r), γ(d/2 − r)) = d − 2r. This shows that 

Hessd is zero in this direction, and we have that 

(2.5) Hessv((E (x) 

n , −E (y) 

n ), (E (x) 

n , −E (y) 

n )) = 0 + 1 

2 ϕ′′ (−2) −2 = 2ϕ ′′ . 

For the other terms, we let σ(r) = (exp x(rE (x) 

i ), exp y(rE (y) 

i )) and define f(r) = d(σ(r)). Notice that 

f(r) ≤ length(γr) where γr is any geodesic between expx(rE (x) 

i ) and expy(rE (y) 

i )). In particular, we 

choose (the choice comes from “transplanting of Jacobi fields” from the constant curvature space) 

 

. 

(2.6) γr(s) = exp γ(s) 

−d/2 

 

rck(s) 

ck(d/2) E(γ(s)) 

i 

Because f(r) lies below length(γr) and is tangent at r = 0, we see that 

f ′ (0) = d 

 

 

length(γr) 

dr r=0 

f ′′ (0) ≤ d2 

dr2 (2.7) 

 

 

length(γr). 

r=0 

By the standard second variation of arc length, we see that 

d 

(2.8) 

2 

dr2 

 

1 

length(γr) = 

r=0 (ck(d/2)) 2 

 

d/2 

d 

ds ck 

2 − (ck) 2 

R(Ei, En, Ei, En) ds. 

Summing from i = 1 to n − 1 and using Ric ≥ (n − 1)k, we get that 

(2.9) 

n−1 

i=1 

Hessd((E (x) 

i , E (y) 

i ), (E (x) 

i , E (y) 

i )) ≤ 

n − 1 

(ck(d/2)) 2 

d/2 

−d/2 

= −2(n − 1)Tk(d/2) 

d 

ds ck 

2 

− (ck) 2 k 

The last equality follows from integrating by parts. Combining the above results, the theorem 

follows (at smooth points of v). 

Now, we recall 

Definition 2.5. We say that L (v) ≤ 2Lϕ in the viscosity sense if for any point (x0, y0) ∈ M × M 

and a C 2 function ψ(x, y) defined near (x0, y0) with 

(1) ψ(x, y) ≤ v(x, y) near (x0, y0) and 

(2) ψ(x0, y0) = v(x0, y0), 

then L ψ ≤ 2Lϕ at (x0, y0). 

However, examining the above proof, it is clear that that this statement holds as well. Thus, we 

have proven the theorem. 

 

ds


Figure 1. The top function does not satisfy u ′′ ≤ 0 in the viscosity sense, while 

the bottom one does. 

We give an example of a viscosity solution to a differential inequality in Figure 1. 

Now, to prove Theorem 2.1, we let ϕ1 be the first eigenfunction of L. It is clear that ϕ ′ 1 

Furthermore, ϕ(s, t) = e −tµ ϕ1(s) solves 

(2.10) 

∂ϕ 

∂t 

= Lϕ 

ϕ|t=0 = ϕ1. 

We further assume that u(x, t) is a solution to the heat equation on M 

(2.11) 

∂u 

∂t 

= ∆u 

u|t=0 = u0. 

As we have remarked previously, we will finish the proof of Theorem 2.1 if we can show 

Claim 2.6. For u and ϕ defined as above, we have that 

 

d(x, y) 

|u(x, t) − u(y, t)| ≤ 2Cϕ , t , 

2 

where C is chosen so that the inequality holds at t = 0. 

Proof. For ɛ > 0, we consider 

 

d(x, y) 

(2.12) Zɛ(x, y, t) = u(y, t) − u(x, t) − 2Cϕ , t − ɛ(1 + t). 

2 

Because Zɛ(x, y, 0) ≤ −ɛ < 0, if Zɛ(x, y, t) > 0 for some (x, y, t), then there is (x0, y0) so that 

Zɛ(x0, y0, t0) = 0 and Zɛ(x, y, t) < 0 for t < t0. We note that x0 = y0, because Zɛ = −ɛ(1 + t) on 

the diagonal. Thus, defining 

(2.13) ψ(x, y) = u(y, t0) − u(x, t0) − ɛ(1 + t0), 

 

we have that ψ(x0, y0) = 2Cϕ 

and ψ(x, y) ≤ 2Cϕ 

 

d(x0,y0) 

2 , t0 

d(x,y) 

2 , t0 

> 0. 

 

for (x, y) near (x0, y0). 

Thus, by (the viscosity part of) Theorem 2.4, we have that L (v) ≤ 2CLϕ. However, because v is 

of product type, by definition of L , we see that 

(2.14) ∆yu(y0, t0) − ∆xu(x0, t0) ≤ 2CLϕ = 2C dϕ 

dt 

Thus 

(2.15) 

∂ 

 

 

dϕ 

(u(y, t) − u(x, t) − ɛ(1 + t)) + ɛ ≤ 2C 

∂t (x0,y0,t0) dt , 

or ∂Zɛ 

∂t (x0, y0, t0) ≤ −ɛ < 0. This is a contradiction, proving our claim.


k = 0 

k < 0 

Figure 2. The manifolds showing sharpness of Theorem 2.1. Notice that the capping 

process does not destroy the eigenvalue bounds or the curvature bounds, in 

particular because the area of the un-capped surfaces is ∼ a, while the area of the 

hemispheres is ∼ a 2 . 

Finally, we note that we have only shown that λ1(n, k, D) ≥ µ(n, k, D). To show equality, we 

must exhibit (M n i , gi) with Ricgi ≥ (n − 1)k and diam(M n i , gi) ≤ D and λ1(M n i , gi) → µ. We 

will briefly discuss the simplest cases k = 0 and k < 0. First, on S n−1 × [−D/2, D/2] we define 

ga = ack(s)g S n−1 + ds 2 for some constant a. One may easily check that (if k ≤ 0), this has 

Ric ≥ (n − 1)k for any a > 0. Furthermore, one may check that ck(s) used as a test function in 

the Rayleigh quotient shows that lim sup a→0 λ D 1 (ga) ≤ µ. Finally, one may cap off the ends with 

hemispheres to get a sequence of closed Riemannian manifolds with the desired properties. These 

manifolds are illustrated in Figure 2. 

2.1. The Fundamental Gap Problem. We briefly remark that the techniques discussed here 

were also used by Andrews-Clutterbuck to prove 

Theorem 2.7 (The Fundamental Gap Conjecture [AC11]). For Ω ⊂ R n a bounded convex domain 

and V a weakly convex potential, define L := ∆ − V . Considering the Dirichlet eigenvalue problem, 

we denote the eigenvalues by 0 < λ1 < λ2 ≤ . . . . Then there is a lower bound on the “fundamental 

gap” 

λ2 − λ1 ≥ 3π2 

D 2 

where D is the diameter of Ω. Furthermore, this is a sharp bound. 

This was a natural conjecture coming from the study of Schrödinger operators, formulated independently 

by van den Berg, Ashbaugh-Benguria and Yau around 1983. Various authors obtained 

non sharp results, e.g. Wong-Yau obtained a lower bound of π2 

4D2 , see [SY94, p. 131] (the paper 

[AC11] gives a detailed list of prior works). 

3. Background on Surfaces and Harmonic Maps 

We will need to recall certain facts about two dimensional geometry. If (M 2 , g) is an oriented 

surface with metric g, then the metric and orientation define isothermal coordinates, i.e. coordinates 

x 1 , x 2 so that gij = λ(x) 2 δij for some λ(x 1 , x2) > 0. Isothermal coordinates always exist. 

Furthermore, given isothermal coordinates, writing z = x 1 + √ −1x 2 , may define a holomorphic 

coordinate patch. One may check that the change of coordinates between such patches is conformal 

and thus holomorphic. 

In particular, we recall 

Theorem 3.1 (Uniformization Theorem). If (M 2 , g) is oriented (but not necessarily complete) so 

that π1(M) = {1} then (M, g) is conformally equivalent to one of 

(1) the sphere S 2 ,


(2) the unit disk D (with either the flat metric or the hyperbolic metric—it is irrelevant since 

both conformal to the other), or 

(3) the complex plane C. 

Corollary 3.2. Every Riemann surface has a compatible conformal metric which is complete and 

has constant curvature. 

3.1. Area and Energy of Maps. Consider F : (M 2 , g) → (N n , h) (which we do not assume to 

be a Riemannian immersion). We define the (unsigned) area of (the image of) F to be 

 

 

det(F 

(3.1) A(F ) = Area(F ) := 

∗h) det(g) dVg. 

Alternatively, choosing coordinates x 1 , x 2 , we may write 

 

(3.2) A(F ) = 

or 

 

(3.3) A(F ) = 

M 

M 

M 

F∗(∂/∂x 1 ) ∧ F∗(∂/∂x 2 )dx 1 ∧ dx 2 

F∗(∂/∂x 1 ) 2 F∗(∂/∂x 2 ) 2 − 〈F∗(∂/∂x 1 ), F∗(∂/∂x 2 )〉dx 1 ∧ dx 2 . 

We emphasize that this is the unsigned area, so in particular if n = 2 then A(F ) = 

We also define the energy of F to be 

 

(3.4) E(F ) = Energy(F ) := trg(F ∗ h)dVg. 

In coordinates x 1 , x 2 , (F ∗ h)ij = F∗(∂/∂x 1 ), F∗(∂/∂x 2 ) , so 

(3.5) e(F ) := trg(F ∗ h) 

We have 

M 

2 

g ij F∗(∂/∂x 1 ), F∗(∂/∂x 2 ) . 

i,j=1 

M 

| Jac(F )|dV . 

Claim 3.3. We have A(F ) ≤ 1 

2E(F ) with equality if and only if F is weakly conformal, i.e. 

F ∗h = λ2g with λ ≥ 0. 

Proof. Let x 1 , x 2 be isothermal coordinates, so gij = u 2 δij. Thus 

(3.6) e(F ) = u −2 

Furthermore, 

2 

F∗(∂/∂x i ) 2 . 

(3.7) dVg = det(g)dx 1 ∧ dx 2 = u 2 dx 1 ∧ dx 2 . 

Combining these two equations shows that 

 

(3.8) E(F ) = 

M 

i=1 

i=1 

2 

F∗(∂/∂x i ) 2 dx 1 ∧ dx 2 .


Thus 

(3.9) 

 

A(F ) = 

 

≤ 

≤ 1 

2 

M 

M 

 

F∗(∂/∂x 1 ) 2 F∗(∂/∂x 2 ) 2 − 〈F∗(∂/∂x 1 ), F∗(∂/∂x 2 )〉dx 1 ∧ dx 2 

F∗(∂/∂x 1 ) 2 F∗(∂/∂x 2 ) 2 dx 1 ∧ dx 2 

M 

i=1 

= 1 

E(F ) 

2 

2 

F∗(∂/∂x i ) 2 dx 1 ∧ dx 2 

In the third line, we used the AM-GM inequality. Finally, to check the case of equality, we see 

that if equality holds in each step, then F∗(∂/∂x 1 ) and F∗(∂/∂x 2 ) must be orthogonal (by the first 

inequality being an equality) and have the same norm (by equality in AM-GM). This easily implies 

that F must be weakly conformal. 

3.2. Harmonic Maps. As above, we take F : (M 2 , g) → (N n , g). We define 

Definition 3.4. The map F is harmonic if F is stationary (i.e a critical point) for E(F ) (under 

variations of the map F ). 

It is readily computed that the Euler-Lagrange equations in isothermal coordinates take the form 

(3.10) ∆F := 

2 

i=1 

∇ ∂ 

∂xi F∗ 

 

∂ 

∂xi 

= 0. 

We remark that the connection used here is really the Levi-Civita connection on (N, h) pulled back 

by F ∗ , i.e. F ∗ (∇ N ), which is a connection on the bundle F ∗ T N → M (clearly this connection ∇ 

is compatible with the metric h on the bundle F ∗ T N). If we also choose coordinates u 1 , . . . , u n on 

N, then writing u α = u ◦ F , we may also write this as 

(3.11) ∆δu α + 

2 

n 

Γ 

i=1 β,γ=1 

α βγ (u(x))∂uβ 

∂xi ∂uγ = 0, 

∂xi for α = 1, . . . , n. Notice that because the energy functional is metric independent, the Euler- 

Lagrange equations are as well (so the Laplacian is taken with respect to the flat metric). To check 

(3.11), we note that the Christoffel symbols of ∇ = F ∗ (∇ N ) may be computed as 

(3.12) 

F ∗ T N Γ γ 

iβ 

∂ 

= ∇ ∂ 

∂uγ = ∇ N 

∂x i 

∂ 

∂u β 

“ 

∂ 

F∗ 

∂xi ” ∂ 

∂uβ = ∂uα 

∂x i ∇N ∂ 

∂u α 

= ∂uα 

∂x i 

N Γ δ α,β 

∂ 

∂uβ ∂ 

, 

∂uγ and thus the desired formula follows from the Leibniz rule for connections. 

We note 

Remark 3.5. The map F is harmonic and conformal if and only if F (M) := Σ is a minimal surface 

in N.


To see that harmonic and conformal implies minimal, choosing isothermal coordinates we have 

that the second fundamental form of Σ is 

(3.13) II Σ 

 

∂ 

F∗ 

∂xi 

∂ 

, F∗ 

∂xj 

= ∇ ∂ 

∂xi 

∂ 

F∗ 

∂xj ⊥Σ 

∂ 

and if F is conformal, F∗ ∂xi 

is an orthogonal basis, so F ∗ (h) = λ2δ. Thus, we have that the 

mean curvature of Σ is 

 

2 

(3.14) 

H Σ = λ −2 

= λ −2 

i=1 

2 

i=1 

= λ −2 (∆F ) ⊥Σ 

 

∇ ∂ 

∂xi 

∂ 

F∗ 

∂xi 

⊥Σ 

∇ ∂ 

∂xi F∗ 

 

∂ 

∂xi ⊥Σ where (·) ⊥Σ denotes projection onto the normal bundle of Σ ↩→ N 3 

On the other hand, if F is a minimal surface, then choosing g = F ∗h, we have made F into 

an isometry (and thus a conformal map). The above computation shows that (∆F ) ⊥Σ = 0, so 

we just need to show that the tangential directions vanish as well. However, this always happens, 

irrespective of the minimality of F . To see this, we compute (the computation for ∂/∂x2 is identical) 

 

∂ 

∆F, F∗ 

∂x1 

= ∇ ∂ 

∂x1 

∂ 

F∗ 

∂x1 

∂ 

, F∗ 

∂x1 

+ ∇ ∂ 

∂x2 

∂ 

F∗ 

∂x2 

∂ 

, F∗ 

∂x1 

= 1 ∂ 

2 ∂x1 

 

 

F∗ 

∂ 

∂x1 

2 

∂ 

− F∗ 

∂x2 

, ∇ ∂ 

∂x2 

∂ 

F∗ 

∂x1 

= 1 ∂ 

2 ∂x1 

 

 

F∗ 

∂ 

∂x1 

2 

∂ 

− F∗ 

∂x2 

, ∇ ∂ 

∂x1 

∂ 

F∗ 

∂x2 

= 1 ∂ 

2 ∂x1 

 

 

F∗ 

∂ 

∂x1 

2 

− 1 ∂ 

2 ∂x1 

 

 

F∗ 

∂ 

∂x2 (3.15) 

 

2 

= 0. 

 

∂ 

The second equality follows from the fact that F is conformal so F∗ ∂x1 

∂ and F∗ ∂x2 

are orthogonal. 

The third equality follows from symmetry of the Hessian and finally, conformality is used to 

show that 

 

 

(3.16) 

 

F∗ 

∂ 

∂x1 

 

= 

F∗ 

∂ 

∂x2 

 

. 

Two important special cases of harmonic maps are when (N, g) is Euclidean space (R n , δ) or the 

round sphere (S n , g). In the first case, we have 

Example 3.6. A map F : (M 2 , g) → (R n , δ) has F (M) minimal if and only if ∆F α = 0, for 

α = 1, . . . , n and F is conformal. 

Example 3.7. In general, a map F : (M 2 , g) → N n ⊂ Rk given in the form F = (F 1 , . . . , F k ) with 

F (p) ∈ N for all p ∈ M has the energy functional 

k 

E(F ) = ∇ N F α 2 dVg 

M α=1 

3 Note that the normal bundle is naturally a subbundle of F ∗ T N, so our conventions are coherent.


and Euler-Lagrange equations (∆F ) N = 0 (i.e. the component which is tangent to N vanishes) 4 . 

In particular, the Euler-Lagrange equations are equivalent to requiring that ∆F is tangent to N. 

In isothermal coordinates, this is equivalent to 

2 

 

∂2F ∆F = 

∂xi∂xi ⊥N = 

i=1 

2 

II N 

i=1 

 

∂F ∂F 

, 

∂xi ∂xi 

, 

where IIN is the (vector valued) second fundamental form of N ⊂ Rk . In particular, if N = Sn ⊂ 

Rn+1 , we have that 

IIx(v, w) = − 〈v, w〉 x, 

so F : M → Sn is harmonic means that 

∆δF α 2 

 

 

+ 

∂F 

∂xi 

 

2 

 

F α = 0. 

i=1 

In particular, if F is conformal as well, then F ∗ (gS n) = λ2 δ, so we easily see that this implies 

∆δF α + 2λ 2 F α = 0, or in other words 

∆gF α + 2F α = 0. 

Thus, we have shown that conformal minimal immersions into a sphere are given in coordinates 

with eigenfunctions with respect to the induced metric. 

Finally, we remark that given F : (M 2 , g) → (N 2 , h) a conformal map, for u a function on N, 

we have that 

(3.17) EM(u ◦ F, g) = EN(u, h) 

and thus ∆hu = 0 implies that ∆g(u ◦ F ) = 0. To check the above formula, we note that F : 

(M, F ∗ (h)) → (N, h) is a (local) isometry, so 

(3.18) EN(u, h) = EM(u ◦ F, F ∗ (h)) 

If we let F ∗ (h) = λ 2 g, then the right hand side is equal to EM(u ◦ F, g) by conformal invariance of 

energy. 

4. Conformal Balancing of Measures 

We will make extensive use of the following result 

Theorem 4.1. Let µ be a probability measure on Bn+1 , the closed unit ball. Assume that µ has no 

point mass on ∂B, i.e. µ({p}) = 0 for all p ∈ ∂B. Then, there is a conformal diffeomorphism ϕ of 

B (extending to a conformal diffeomorphism of the boundary) so that 

 

(4.1) 

ϕ(x)dµ(x). 

B n+1 

Proof. Associated to any y ∈ Bn+1 , we will choose ϕy : Bn+1 → Bn+1 , a conformal diffeomorphism 

(extending to the boundary) with the following three properties: (1) ϕ0 = idB, (2) ϕy(−y) = 0 and 

(3) ϕy is a conformal dilation fixing the two points ± y 

|y| . It is not hard to see that this uniquely 

determines the map ϕy, and an explicit formula is 

 

1 − |y| 2 

∗ 

(4.2) ϕy(x) = (x + y) + y 

|x + y| 2 

where z∗ = z 

|z| 2 . Alternatively, one may describe ϕy geometrically as the map which is an conformal 

inversion in a sphere S−y which is chosen as in Figure 3 to meet ∂Bn+1 orthogonally and at the 

4 We emphasize that, contrary to the computations above, we are not projecting onto the tangent and normal 

bundle to N ↩→ R k , rather than Σ ↩→ N


points of intersection between the plane normal to −y and ∂B n+1 , followed by an inversion with 

respect to Sy. 

Sy 

∂B n+1 

y 

|y| 

y 

Figure 3. The map ϕy is simply an inversion with respect to the sphere S−y which 

is the unique sphere orthogonal to ∂B n+1 at the points shown (i.e. the points on 

∂B n+1 ∩ P where P is the plane through −y which is normal to y), followed by an 

inversion with respect to Sy. 

Now, we consider the function 

 

(4.3) F (y) = 

B n+1 

−y 

- y 

|y| 

ϕy(x)dµ(x). 

It is not hard to check that because there are no point masses, F extends continuously to B n+1 and 

F |∂B = id∂B. The desired conclusion thus follows from a fixed point argument. 

4.1. First Eigenvalue Bounds by Conformal Balancing. Here, we discuss several classical 

eigenvalue bounds using Theorem 4.1, the conformal balancing theorem. This first result is due to 

J. Hersch in 1970. 

Theorem 4.2 ([Her70]). For (M 2 , g) a sphere with an arbitrary metric, we have that 

λ1(M 2 , g) Area(M 2 , g) ≤ 8π 

with equality if and only (M 2 , g) is a round sphere. 

Proof. By the uniformization theorem, there is a conformal map to the round sphere F : (M 2 , g) → 

(S2 , g0) ⊂ R3 . Notice that thus, F ∗ (g0) = 1 

1 

2e(F )g = 2 |∇F |2g. We’ll consider µ := F#(µg), the 

pushforward under the map F of the g-volume measure. By the conformal balancing theorem, 

there is ϕ : B3 → B3 so that 

 

(4.4) 

ϕ dµ = 0. 

We may rewrite this as 

(4.5) 

 

M 2 

S 2 

ϕ ◦ F dµg = 0, 

S−y


and because ϕ ◦ F is also a conformal map ϕ ◦ F : (M 2 , g) → (S2 , g0), we may assume without loss 

of generality that we actually chose F so that 

 

(4.6) 

F dµg = 0. 

M 2 

Thus, the components of F = (F 1 , F 2 , F 3 ) are orthogonal to the constant functions on M, and are 

thus acceptable test functions for λ1(g). As such, we have that 

 

(4.7) λ1(g) (F i ) 2 

dµg ≤ |∇F i | 2 dµg. 

M 

Summing over i = 1, 2, 3, we thus have that (because |F | = 1) 

 

(4.8) λ1(g) Area(g) ≤ |∇F | 2 

dµg = e(F ) dµg = 2A(F ) = 8π. 

M 

We used the fact that F is conformal in the first equality. 

In the case of equality, we see that the F i ’s are first eigenfunctions. By scaling g, we assume 

(4.9) ∆gF i = −2F i . 

Finally, because |F | = 1, 

(4.10) 

M 

M 

0 = 1 

∆|F |2 

2 

= |∇F | 2 + ∆F · F 

= |∇F | 2 − 2|F | 2 

= |∇F | 2 − 2. 

Thus, we see that F ∗ (g0) = 1 

2 |∇F |2 g = g, so F is an isometry. 

The argument used by Hersch is similar to the argument used by Weinstock in 1954 to prove 

Theorem 4.3 ([Wei54]). For Ω ⊂ R 2 simply connected, then 

σ1(Ω)L(∂Ω) ≤ 2π 

where σ1 is the first (nonzero) Steklov eigenvalue (as defined in Definition 1.3). Furthermore, 

equality holds if and only if Ω is a round disk. 

Proof. By the Riemann mapping theorem, there is a conformal map F : Ω → D. 5 Letting ds denote 

the arc length measure of ∂Ω, by applying the conformal balancing argument as above, we may 

assume without loss of generality that 

 

(4.11) 

F ds = 0. 

∂Ω 

We briefly recall that the variational characterization of the first nonzero Steklov eigenvalue is 

 

Ω (4.12) σ1(Ω) = inf 

|∇u|2 

 

: u = 0 . 

This shows that 

 

(4.13) σ1(Ω) 

∂Ω 

∂Ω u2 

(F i ) 2 

≤ 

Ω 

∂Ω 

|∇F i | 2 

5 The possibility of the target being C is precluded by the Liouville theorem, because there are clearly nonconstant 

harmonic functions on Ω, obtained by solving a Dirichlet problem.


Summing for i = 1, 2 gives 

 

(4.14) σ1(Ω)L(∂Ω) ≤ 

Ω 

|∇F | 2 = 2 Area(D) = 2π. 

The equality case is slightly different. If equality holds, we may assume that the F i are eigenfunctions 

with eigenvalue 1. Thus ∆F i ∂F i 

= 0 in Ω and ∂ν = F i on ∂Ω. The second fact shows that 

∂F 

∂ν is a unit vector on ∂Ω. Thus, by conformality 

1 

(4.15) 

2 |∇F |2 

 

= 

∂F 2 

 

∂ν = 1 

on ∂Ω. However, we would like this on all of Ω. To see this, we apply the maximum principle to 

 

1 

(4.16) ∆ log |∇F |2 = 0, 

2 

which holds because the log of the modulus of a holomorphic function is always harmonic. 

We remark that if we try to generalize this to surfaces with boundary, (M 2 , g), if we let ˆg = λ 2 g 

where λ > 0 and λ ≡ 1 on ∂M, then σk(M, g) = σk(M, ˆg). In particular, we define 

Definition 4.4. Two surfaces with boundary (M1, g1) and (M2, g2) are σ-isometric if there is a 

conformal map F : (M1, g1) → (M2, g2) with 1 

2 |∇F |2 = 1. 

Clearly in this case, σk(M1, g1) = σk(M2, g2). We may define a slightly more general notion 

Definition 4.5. Two surfaces with boundary (M1, g1) and (M2, g2) are σ-homothetic if there is a 

conformal map F : (M1, g1) → (M2, g2) with 1 

2 |∇F |2 = α > 0 on ∂M, where α is some constant. 

Again, under this condition, σk(M1, g1)L(∂M1, g1) = σk(M2, g2)L(∂M2, g2). In particular, using 

this definition allows us to generalize (the equality case of) Weinstock’s theorem to surfaces 

Theorem 4.6. For (M 2 , g) a simply connected surface, 

σ1(M)L(∂M) ≤ 2π 

with equality if and only if (M, g) is σ-homothetic to D. 

The proof is identical, except we in the equality case all we may conclude is that |∇F | 2 is constant 

on ∂M, and thus we have the σ-homothetic property by the same argument as before. 

The following theorem due to Szegő (also in 1954) is also proven with a similar, although more 

involved, argument, so we discuss it here 

Theorem 4.7 ([Sze54]). For Ω ⊂ R 2 a simply connected planar domain, we have a bound on the 

first nonzero Neumann eigenvalue of Ω as follows 6 

We remark that the right hand side is ≈ 3.4π. 

λ N 1 (Ω) Area(Ω) ≤ λ N 1 (D) Area(D). 

We remark that on the upper half-sphere S 2 + = S 2 ∩ {z ≥ 0}, the functions x, y restricted to S 2 + 

are easily seen to be the first Neumann eigenfunctions, with eigenvalue 2. Thus µ1(S 2 +) Area(S 2 +) = 

4π > 3.4π, so this bound cannot generalize to arbitrary surfaces with boundary. 

Proof. We recall that by separation of variables, the first Neumann eigenfunctions on D are given 

by f(r) cos θ and f(r) sin θ where f(r) is a Bessel function solving the ODE 

(4.17) r 2 f ′′ + rf ′ + (r 2 − 1)f = 0 

6 One may also characterize the case of equality, but we will not discuss this here


with boundary condition f ′ (1) = 0, and because we require this to correspond to a smooth eigenfunction 

at the origin f ′ (0) = 0. We will later need that (f(r) 2 ) ′ ≥ 0 for r ∈ [0, 1]. To check this, 

suppose otherwise. Then, it attains a negative minimum at some point r0 ∈ (0, 1). Thus, at r0, 

0 = (f 2 ) ′′ = 2(f ′ ) 2 + 2f ′′ f. Thus at r0, 

(4.18) (f ′ ) 2 = −f ′′ f = 

f ′ 

r0f + (r0) 2 − 1 

r 2 0 

However, this is a contradiction, because both terms on the right hand side are strictly negative. 

We define 

(4.19) H : D → R 2 

(r, θ) ↦→ (f(r) cos θ, f(r) sin θ). 

Let F : Ω → D be a conformal map. We claim that by balancing, we may produce a conformal 

diffeomorphism ϕ : D → D so that if ψ := H ◦ ϕ, then 

 

(4.20) 

ψ d(F#µ0) = 0, 

D 

where µ0 is Lebesgue measure on Ω. This is not exactly in the form we proved the balancing 

theorem for, but it is clear that the proof also works for this statement. Thus, we may assume that 

the conformal map F was chosen so that 

 

(4.21) 

H ◦ F dµ0 = 0. 

Ω 

Now, we fix Ψ = H ◦ F . The coefficients are thus admissible test functions for the first Neumann 

eigenvalue, so we have that 

(4.22) λ N 

1 (Ω) (Ψ i ) 2 

≤ |∇Ψ i | 2 . 

Ω 

Summing over i = 1, 2 and using the facts that |H| 2 = f(r) 2 and that the second integral is then 

conformally invariant 

λ N 

1 (Ω) (f ◦ F ) 

Ω 

2 

≤ |∇H| 

D 

2 

= λ N 

1 (D) |H| 

D 

2 

= λ N 

1 (D) f 2 (4.23) 

. 

On the other hand, 

(4.24) 

 

Ω 

(f ◦ F ) 2 

= 

D 

Ω 

D 

f 2 | Jac(F −1 )|. 

We may choose coordinates so that F −1 : D → D is given by w = w(z), and thus this is equal to 

 

(4.25) 

f 

D 

2 

 

 

 

∂w 

 

∂z = 

1 

f(r) 

0 

2 2π 

r 

∂w 

 

 

0 ∂z (reiθ )dθ dr. 

 

:=G(r) 

We claim that 

∂w 

∂z is subharmonic. To see this, notice that because w is holomorphic 

(4.26) 4 ∂2 

 

 

 

∂w 2 

 

 

 

∂z∂z ∂z = 4 

∂ 

 

2w ∂z2 2 

 

 

≥ 0.


By the mean-value property for subharmonic functions, we have that G(r) is non-decreasing with 

r. Thus 

(4.27) 

We thus compute 

(4.28) 

1 

0 

r 

0 

f(r) 2 rG(r)dr = 

G(t)tdt = r 2 

1 

0 

d 

dr 

= f(1) 2 

≥ 

= 

1 

0 

1 

0 

1 

0 

r 

1 

0 

= Area(Ω) 1 

π 

0 

G(sr)s ds ≤ r 2 

 

G(t)t dt 

G(t)t dt − 

 

G(t)t dt f(1) 2 − 

 

G(t)t dt 2 

 

|H| 

D 

2 

1 

0 

G(t)t dt. 

1 

(f(r) 

0 

2 ) ′ 

r 

0 

1 

r 

0 

2 (f(r) 2 ) ′ dr 

1 

0 

rf(r) 2 dr 

 

G(t)t dt dr 

We have used (4.27) in the third line (combined with the fact proved earlier that (f(r) 2 ) ′ ≥ 0), and 

in the final line, we have used the fact that f(r) 2 = |H| 2 . Thus, combining this with (4.23), yields 

the desired result. 

We remark that the key here was that 

∂w 

∂z was subharmonic. By a Bochner formula argument 

(cf. [ES64]), one can show that this also holds when Kg ≤ 0, and in particular the same argument 

as above yields 

Corollary 4.8. For (M 2 , g) with Kg ≤ 0, 

λ N 1 (M, g) Area(M, g) ≤ λ N 1 (D) Area(D). 

4.2. Coarse Upper Bounds By Conformal Balancing. Here we discuss a generalization of the 

conformal balancing argument to bound the first eigenvalue of surfaces with higher genus and/or 

number of boundary components. The inequalities will not be sharp (or are not expected to be 

sharp) in most cases. The first statement in the following theorem is due to Yang-Yau [YY80] and 

the first term in the second statement is due to [FS11] 

Theorem 4.9. For M 2 an orientable surface, letting γ denote the genus of M. Then, for any 

metric g, we have the eigenvalue bounds 

(1) If ∂M = ∅, then 

 

γ + 3 

λ1(M, g) Area(M, g) ≤ 8π . 

2 

(2) If ∂M = ∅, then letting k = |π0(∂M)| denote the number of boundary components, we have 

 

 

γ + 3 

σ1(M, g)L(∂M, g) ≤ min 2π(γ + k), 8π . 

2 

We remark that if γ = 0 in the first statement, we obtain the Hersch bound, Theorem 4.2, but 

for higher genus it is not expected to be sharp. We also remark that a surface of genus γ and with 

k boundary components is homeomorphic to a genus k surface with k disks removed. 

Proof. To prove statement (1), we choose a conformal branched cover of the round sphere by M, 

F : M → S 2 ⊂ R 3 . If the genus γ > 0, then certainly deg(F ) > 1. By conformal balancing, we


may assume that 

(4.29) 

 

M 

F = 0. 

Using the components of F as test functions in the Rayleigh quotient and summing yields 

 

(4.30) λ1(M, g) |F | 2 

≤ |∇F | 2 . 

M 

Because the right hand side is equal to 2 Area(F ) and this is equal to 8π deg(F ), since we are 

considering unsigned area, and almost all points of S 2 are covered deg(F ) times. Thus, we have 

(4.31) λ1(M, g) Area(M, g) ≤ 8π deg(F ). 

 

γ+3 

As such, the theorem will follow assuming we can find a cover F with deg(F ) ≤ 2 . This is the 

subject of “Brill-Noether theory,” cf. [GH94, p. 261] and [Yan94] 

We now will prove the first bound in item (2) of the statement of the theorem, i.e. that 

σ1(M, g)L(∂M, g) ≤ 2π(k + γ). A crucial ingredient in the proof is the following result of Ahlfors. 

The bound was recently improved by Gabard, so we state this version here 

Theorem 4.10 ([Ahl50, Gab06]). For (M 2 , g) a surface of genus γ and with k boundary components, 

then there is a conformal branched cover F : M → S 2 with deg(F ) ≤ γ + k. 

We fix such a cover F : M → S2 and by balancing pushforward of the ∂M boundary measure 

µ0 we may assume that 

 

(4.32) 

F dµ0 = 0. 

We may thus apply the variational characterization of eigenvalues to conclude that 

(4.33) σ1(M, g)L(∂M, g) ≤ 2 Area(F ) = 2π deg(F ) ≤ 2π(k + γ). 

To demonstrate the other bound σ1(M, g)L(∂M, g) ≤ 8π 

nents of M with k disks, letting 

(4.34) 

∂M 

M 

ˆ M = M ∪ D ∪ · · · ∪ D. 

γ+3 

2 

 

, we fill in the boundary compo- 

We may extend the metric g into the disks in an arbitrary manner so that ˆg is at least Lipschitz. 

Letting µ0 be the measure on ˆ M coming from the boundary measure of ∂M with respect to g, as 

in the proof of (1), we may find a conformal branched cover F : M → S2 

with deg(F ) ≤ , 

but in this case we balance F#µ0, so we may assume that F : ˆ M → S2 has 

 

(4.35) 

F dµ0 = 0. 

Restricting F to M, the variational characterization argument yields 

(4.36) σ1(M, g)L(∂M, g) ≤ 2 Area(F |M) ≤ 2 Area(F ) = 4π deg(F ), 

∂M 

yielding the desired bound. 

4.3. Conformal Volume. We will first need the following result about the conformal group of 

S n , which says that other than isometries, the maps ϕy as described in Theorem 4.1 are the only 

conformal diffeomorphisms. 

Proposition 4.11 (Liouville Theorem). For n ≥ 3, if F : S n → S n a conformal diffeomorphism, 

then F = R ◦ ϕy for some y ∈ B n+1 and R some rotation in SO(n). 

γ+3 

2


A proof of this may be found in [SY94, pp. 232-3], where the theorem is proven by considering 

the conformal change of Ricci curvature, which yields a PDE constraint on the conformal factor of 

such a map. We let G(n) denote the conformal group of S n . 

Definition 4.12 (Conformal Volume [LY82]). Suppose that ϕ : M 2 → S n is (weakly) conformal. 

We define the conformal volume, due to Li-Yau 

Volc(n, ϕ) := sup Area(f ◦ ϕ). 

f∈G 

and 

Volc(n, M) := inf Volc(n, ϕ), 

ϕ 

where the infimum is taken over ϕ as above. 

Using this, we have 

Theorem 4.13 ([LY82]). For (M 2 , g) a surface without boundary, we have 

λ1(M, g) Area(M, g) ≤ 2 Volc(n, M) 

with equality if and only if (after rescaling so that λ1 = 2) there is a isometric minimal immersion 

by first eigenfunctions ϕ : M → S n . 

Proof. For now, fix a conformal map ϕ : M → Sn . By balancing, we may assume that 

 

(4.37) 

ϕ = 0 

and thus 

(4.38) λ1(M, g) Area(M, g) ≤ 2 Area(ϕ) ≤ 2 Volc(n, ϕ). 

M 

Taking the infimum over such ϕ the inequality follows. 

For the equality statement, assume that λ1(M, g) = 2. Thus we have that Area(M, g) = 

Volc(n, M). Thus, there is a sequence of conformal maps ϕj : M → S n so that 

M ϕj = 0 and 

(4.39) lim 

j→∞ Volc(n, ϕj) = Volc(n, M) = Area(M, g). 

Thus 

 

(4.40) 2 Volc(n, ϕj) ≥ 2 Area(ϕj) = |∇ϕj| 

M 

2 

≥ 2 |ϕj| 

M 

2 = 2 Area(g). 

As such, we see that ϕj is bounded in W 1,2 (M, Sn ). Extracting a subsequence, we may assume 

that ϕj → ϕ weakly in W 1,2 , strongly in L2 and pointwise a.e. We claim that ϕ : M → Sn is an 

isometric immersion by eigenfunctions. 

First, recall that to show that ϕj converges strongly to ϕ it is enough to show that the norms 

converge. Because 

 

(4.41) Volc(n, ϕj) ≥ |∇ϕj| 2 . 

The left hand side converges to 2 Area(g), and by lower semicontinuity of the norm under weak 

convergence, we have that 

 

(4.42) 2 Area(M, g) ≥ |∇ϕ| 2 . 

However, because ϕ converges in L2 , we have that 

M ϕ = 0 and thus we may use the Poincaré 

inequality giving 

 

(4.43) 

|∇ϕ| 

M 

2 

≥ 2 |ϕ| 

M 

2 = 2 Area(M, g). 

M 

M


Thus, 

M |∇ϕ|2 = 2 Area(M, g), so there cannot be energy loss. Thus the ϕj converge in W 1,2 . 

Due to this convergence, we have that 

 

(4.44) 

|∇ϕ| 

M 

2 

= 2 |ϕ| 

M 

2 

which shows that the components of ϕi are eigenfunctions with eigenvalue 2. Finally, using 1 = |ϕ| 2 , 

we have that 

(4.45) 0 = 1 

2 ∆|ϕ|2 = |∇ϕ| 2 + ϕ · ∆ϕ = |∇ϕ| 2 − 2 

Thus |∇ϕ| 2 = 2, and because ϕ is conformal (by the W 1,2 convergence of the ϕj) we have that ϕ 

is an isometry. 

We now discuss the following theorem due to Li-Yau [LY82]. The rigidity statement is due to 

Montiel-Ros [MR85]. 

Theorem 4.14. If ϕ : M 2 → S n is conformal and minimal then Area(ϕ) = Volc(n, ϕ), i.e. 

Area(f ◦ ϕ) ≤ Area(ϕ) for f ∈ G(n). Furthermore, equality holds if and only if f is an isometry, 

as long as n ≥ 3 and ϕ(M) ⊆ S n−1 ⊂ S n . 

Here, we will only prove the equality Area(ϕ) = Volc(n, ϕ), and not the rigidity statement. 

Proof of Theorem 4.14. The proof follows readily from properties of the Willmore functional, so 

we define it and briefly discuss its properties. The crucial observation is that if ˚ A is the trace free 

second fundamental form for Σ := ϕ(M) ⊂ Sn , then ˚ A2 gd Areag|Σ is invariant under conformal 

changes of the background metric g on Sn . Thus, defining 

Definition 4.15 (Willmore Functional). For Σ ↩→ Sn , we define 

 

1 

W (Σ) := 

4 ˚ A 2 

+ K d Area . 

where K is the Gaussian curvature of Σ. 

Σ 

Because the second term is a topological invariant by Gauss-Bonnet, the Willmore function is 

thus conformally invariant. We note that W (S2 ) = 4π. On the other hand, using the Gauss 

equations, one may easily check that 

(4.46) 

1 

4 ˚ A 2 + K = H 2 + 1, 

where H is the mean curvature of Σ. Thus, we have 

(4.47) W (Σ) ≥ Area(Σ), 

with equality if and only if Σ is minimal. 

Thus, for any f ∈ G(n), we have that 

(4.48) Area(f ◦ ϕ) ≤ W (f ◦ ϕ) = W (ϕ) = Area(ϕ), 

as desired. 

Corollary 4.16. If ϕ : M 2 → S n is an isometric and minimal embedding by first eigenfunctions, 

then Vol(M) = Volc(n, M). 

Proof. We may assume that λ1(M, g) = 2. By the Li-Yau eigenvalue estimate, Theorem 4.13, 

Area(M) ≤ Volc(n, M). On the other hand, by definition Volc(n, M) ≤ Volc(n, ϕ) = Area(M) by 

the previous theorem. 

Thus, we may compute the conformal volume in a few cases 

Example 4.17. The simplest example is Volc(S 2 , n) = 4π for n ≥ 2.


Example 4.18. Another example is given by the projective plane, where we have Volc(RP 2 , n) = 

6π for n ≥ 4. To see this, we recall the Veronese embedding. First, we embed ϕ : S 2 → S 4 ⊂ R 5 by 

second eigenfunctions 

ϕ(x, y, z) = 

 

xy, xz, yz, 1 

2 (x2 − y 2 ), 1 

2 √ 3 (x2 + y 2 − 2z 2 

) . 

One may check that |ϕ| = 1 and that this map is conformal with |∇ϕ| 2 constant, so it is an embedding. 

On the other hand it is manifestly harmonic, because the coordinates are all eigenfunctions 

on S 2 . Because it is even, the map descends to RP 2 as a minimal embedding by first eigenfunctions, 

so the conclusion easily follows the fact that then 2 Volc(RP 2 , n) = λ1(RP 2 , n) Area(RP n ) = 6 · 2π. 

In particular, the second example shows that we have solved the Hersch problem on RP 2 , because 

Theorem 4.13 then gives a sharp estimate for the first eigenvalue. 

We can also compute the conformal volume for certain conformal classes of tori, starting with 

Example 4.19 (Square Torus). The Clifford torus is a square torus S 1 × S 1 inside of S 3 defined 

by ϕ(x, y) = (e 2πix , e 2πiy ) ∈ S 3 ⊂ C 2 . One may check that this is an conformal minimal embedding 

by first eigenfunctions and Area(ϕ) = 2π 2 . Thus, we see that Volc(square torus) = 2π 2 . 

Example 4.20 (Rhombic Torus). The 60◦ rhombic torus is the conformal class containing the flat 

torus R2 

/ (1, 0), ( 1 

2 , 

√ 

3 

2 ) 

 

. There is an conformal minimal embedding into S5 ⊂ C3 is given by 

ϕ(x, y) = 1 

 

√ e 

3 

4πi y √ 

3 , e 2πi 

“ 

x− y ” 

√ 

3 , e 2πi 

“ 

x+ y ” 

√ 

3 

One may check that Area(ϕ) = 4π2 

√ 3 > 2π 2 , so Volc(rhombic torus) = 4π2 

√ 3 . 

In fact, as we’ll hopefully discuss later, for any metric g in any conformal class on T 2 , we have 

that λ1(g) Area(g) ≤ 2 4π2 √ , which is sharp by the above example. 

3 

Recall that for Σ 2 ⊂ S n , we have that W (Σ) ≥ Volc(n, Σ), where W (·) is the Willmore functional 

as defined in Definition 4.15. In particular, we have 

Conjecture 4.21 (Willmore Conjecture). For Σ 2 ⊂ S n is homeomorphic to a torus, W (Σ) ≥ 2π 2 . 

This has been recently proven by Marques-Neves when n = 3 [MN12]. In fact, as part of their 

proof they have shown that for Σ ⊂ S 3 of genus > 0, then letting Σt := {p : d(p, Σ) = t} for 

t ∈ [−π, π], then 

(4.49) 2π 2 ≤ sup 

t∈[−π,π] 

Volc(Σt) ≤ W (Σ). 

We now the recall Theorem 4.14, which says that for ϕ : M 2 → S n which is conformal and 

minimal, then for all F ∈ G, A(ϕ) ≥ A(F ◦ ϕ) with equality only if F is an isometry (assuming that 

n ≥ 3 and ϕ(M) ⊆ S n−1 ⊂ S n ). This rigidity statement will be crucial in the following theorem 

Theorem 4.22. Assume that ϕ : M 2 → S n is a conformal minimal embedding by first eigenfunctions 

(and that n ≥ 3 and ϕ(M) ⊆ S n−1 ⊂ S n ). Then, every conformal automorphism of M is an 

isometry. 

Proof. Suppose we have some f : M → M which is a conformal diffeomorphism. We consider 

ϕ ◦ f : M → Sn , which is still a conformal minimal embedding. We claim that the components of 

the map ϕ ◦ f are first eigenfunctions. To see this, by conformal balancing, there is F ∈ G so that 

 

(4.50) 

F ◦ ϕ ◦ f = 0 

M


and thus 

(4.51) 

2 Volc(n, M) = 2 Area(ϕ) = 2 Area(ϕ ◦ f) 

≥ 2 Area(F ◦ ϕ ◦ f) = E(F ◦ ϕ ◦ f) 

≥ λ1(g) Area(M, g) = 2 Volc(n, M). 

The second inequality follows because we have balanced ϕ ◦ f, so the components of F ◦ ϕ ◦ f are 

admissible test functions in for the first eigenvalue. Furthermore, by Theorem 4.14. equality in the 

first inequality holds if and only if F is an isometry. Thus, because equality must hold, we know 

that F is an isometry, and thus the components of ϕ ◦ f are first eigenfunctions. 

Now, by the same argument as in the end of the proof of Theorem 4.13, this readily implies that 

f is an isometry. 

5. Free Boundary Minimal Surfaces and Steklov Eigenvalues 

Now, we discuss related estimates for surfaces with boundary. It will be crucial for us to discuss 

free boundary minimal submanifolds of the ball B n . The simplest example of a free boundary 

minimal submanifold is the cone over a minimal submanifold of S n−1 , as in Figure 4. The crucial 

ν = x 

S n−1 

C(Σ k ) 

Figure 4. The cone over Σ k , a minimal submanifold of S n−1 is an example of a 

free boundary minimal surface. 

observation is that in this case, x 1 , . . . , x n are Steklov eigenvalues because ∆x i = 0 in C(Σ k ) and 

Dνx = x. In general, note that this holds for 

Definition 5.1 (Free Boundary Minimal Submanifold of B n ). A minimal submanifold of B n is a 

free boundary minimal submanifold if its boundary lies on S n−1 and it meets S n−1 orthogonally. 

In particular, just as in the boundaryless case, if M is a surface with nonempty boundary ∂M, 

then for ϕ : M → B n (for n ≥ 3) a proper 7 , conformal map, we can define the conformal volume of 

ϕ to be 

(5.1) Volc(n, ϕ) = sup Area(F ◦ ϕ). 

F ∈G 

We remark that the conformal group of B n is the same as that of the ball. In particular, by using 

the usual balancing arguments we have discussed above, one may show 

7 Note that in this case, this is just the requirement that exactly ∂M is mapped to S n−1 . 

Σ k


Theorem 5.2 (cf. [FS11, Corollary 6.3]). For ϕ : M → S n as above, we have that 

σ1(g)L(∂M, g) ≤ Volc(n, ϕ) 

with equality if and only if ϕ is a free boundary minimal surface which is isometric (homothetic) 

on ∂M. 

This suggests the following question in the spirit of Theorem 4.14 

Question 5.3. If Σ is a free boundary minimal surface and F ∈ G, is it true that |F (Σ)| ≤ |Σ|? 

This seems unlikely to be true, but the following seems more likely 

Conjecture 5.4 (Schoen). If Σ k ⊂ B n is a free boundary minimal surface and F : B n → B n is a 

conformal diffeomorphism of the ball, then |F (∂Σ)| ≤ |∂Σ|. 

5.1. Examples of Steklov-Critical Submanifolds. We can describe several examples of submanifolds 

of B n which maximize the first Steklov eigenvalue in the sense of Theorems 4.9 and 5.2. 

Let M be of genus γ and having k boundary components. We define 

(5.2) σ ∗ (γ, k) := sup{σ1(g)L(∂M, g) : g a smooth metric on M}. 

By Theorem 4.9, this is a finite quantity. A natural question is 

Question 5.5. Is σ ∗ (γ, k) achieved by a smooth metric? If so, by what? 

The simplest case in which we may answer this affirmatively is 

Example 5.6. The flat disk achieves σ ∗ (0, 1) = 2π. 

The next case which the maximizer is understood is 

Example 5.7 (The Critical Catenoid). When γ = 0, k = 2, σ(0, 2) ∗ is achieved by the (unique 

up to rotations) catenoid in B 3 which meets S 2 orthogonally. See Figure 5. Recall that a catenoid 

Figure 5. The critical catenoid, a free boundary minimal submanifold of B 3 . 

may be described by rotating x = cosh z around the z-axis. In particular, the piece in B 3 may be 

parametrized by 

ϕ : [−T, T ] × S 1 → B 3 

ϕ(t, θ) = (cosh t cos θ, cosh t sin θ, t). 

Here, T is a uniquely determined by the requirement that the catenoid meet S 2 orthogonally. One 

may check that this is a conformal harmonic map consisting of Steklov first eigenfunctions.


We also may describe 

Example 5.8 (The Critical Möbius Strip). While we have mostly focused on orientable surfaces in 

the above discussion, we also remark that for the Möbius band R × S 1 /(t, θ) ∼ (−t, θ + π), defining 

the map 

(5.3) ϕ(t, θ) = (2 sinh t cos θ, 2 sinh t sin θ, cosh 2t cos 2θ, cosh 2t, sin 2θ), 

there is a unique T0 so that ϕ| [−T0,T0]×S 1 is a free boundary minimal embedding in to B4 r for some 

radius r (we may then rescale the entire picture so that r = 1). 

It seems that one should be able to use gluing techniques to desingularize the union of the critical 

catenoid and a plane, to obtain free boundary minimal surfaces with three boundary components 

and of high genus, but this has not been carried out. 

Finally, we remark that Fraser-Schoen have constructed explicit maximizers of σ(0, k) [FS11]. 

Which look like two planes joined by necks on the boundary, which increase the number of boundary 

components but not the genus. Their construction shows that σ(0, k) ∗ is strictly increasing with k 

and limk→∞ σ(0, k) ∗ = 4π. 

5.2. Structure of Extremal Metrics. In fact, it turns out that maximizers of the scale invariant 

first eigenfunction must be minimally embedded by first eigenfunctions, i.e. 

Theorem 5.9. For (M 2 , g) with ∂M = ∅, if λ1(g) Area(g) = λ ∗ (M) = sup{λ1(g) Area(g)}, then 

there are ϕ1, . . . , ϕn linearly independent first eigenfunctions so that they combine to give a map 

ϕ : M → S n−1 which is a homothety and a conformal minimal embedding. 

For the boundary case, we have the similar 

Theorem 5.10. For (M 2 , g) with ∂M = ∅, if σ1(g)L(∂M, g) = σ ∗ (M) = sup{σ1(g)L(∂M, g)}, 

then there are ϕ1, . . . , ϕn first Steklov eigenfunctions so that they give a map ϕ : M → B n which 

is conformal and harmonic and ϕ(M) is a free boundary minimal surface. Furthermore, g is σhomothetic 

to ϕ ∗ (δ). 

References 

[AC11] Ben Andrews and Julie Clutterbuck, Proof of the fundamental gap conjecture, J. Amer. Math. Soc. 24 (2011), 

no. 3, 899–916. MR 2784332 (2012d:35051) 

[AC12] Ben Andrews and Julie Clutterbuck, Sharp modulus of continuity for parabolic equations on manifolds and 

lower bounds for the first eigenvalue, http://arxiv.org/abs/1204.5079 (2012). 

[Ahl50] Lars L. Ahlfors, Open Riemann surfaces and extremal problems on compact subregions, Comment. Math. 

Helv. 24 (1950), 100–134. MR 0036318 (12,90b) 

[AN11] Ben Andrews and Lei Ni, Eigenvalue comparison on bakry-emery manifolds, http://arxiv.org/abs/1111. 

4967 (2011). 

[BQ00] Dominique Bakry and Zhongmin Qian, Some new results on eigenvectors via dimension, diameter, and Ricci 

curvature, Adv. Math. 155 (2000), no. 1, 98–153. MR 1789850 (2002g:58048) 

[CL81] Shiu Yuen Cheng and Peter Li, Heat kernel estimates and lower bound of eigenvalues, Comment. Math. 

Helv. 56 (1981), no. 3, 327–338. MR 639355 (83b:58076) 

[Cro80] Christopher B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. 

(4) 13 (1980), no. 4, 419–435. MR 608287 (83d:58068) 

[ES64] James Eells, Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 

(1964), 109–160. MR 0164306 (29 #1603) 

[FS11] Ailana Fraser and Richard Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces, 

Adv. Math. 226 (2011), no. 5, 4011–4030. MR 2770439 (2012f:58054) 

[Gab06] Alexandre Gabard, Sur la représentation conforme des surfaces de Riemann à bord et une caractérisation 

des courbes séparantes, Comment. Math. Helv. 81 (2006), no. 4, 945–964. MR 2271230 (2007k:14122) 

[GH94] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & 

Sons Inc., New York, 1994, Reprint of the 1978 original. MR 1288523 (95d:14001) 

[Her70] Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. 

Paris Sér. A-B 270 (1970), A1645–A1648. MR 0292357 (45 #1444)


[LY82] Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and 

the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269–291. MR 674407 (84f:53049) 

[MN12] Fernando C. Marques and André Neves, Min-max theory and the willmore conjecture, http://arxiv.org/ 

abs/1202.6036 (2012). 

[MR85] Sebastián Montiel and Antonio Ros, Minimal immersions of surfaces by the first eigenfunctions and conformal 

area, Invent. Math. 83 (1985), no. 1, 153–166. MR 813585 (87d:53109) 

[SY94] R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in 

Geometry and Topology, I, International Press, Cambridge, MA, 1994, Lecture notes prepared by Wei Yue 

Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu, Translated from the Chinese 

by Ding and S. Y. Cheng, Preface translated from the Chinese by Kaising Tso. MR 1333601 (97d:53001) 

[Sze54] G. Szegő, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 

343–356. MR 0061749 (15,877c) 

[Wei54] Robert Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal. 3 (1954), 745–753. 

MR 0064989 (16,368c) 

[Yan94] Kichoon Yang, Meromorphic functions, divisors, and projective curves: an introductory survey, J. Korean 

Math. Soc. 31 (1994), no. 4, 569–608. MR 1312529 (96a:14031) 

[YY80] Paul C. Yang and Shing Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal 

submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 1, 55–63. MR 577325 (81m:58084) 

[ZY84] Jia Qing Zhong and Hong Cang Yang, On the estimate of the first eigenvalue of a compact Riemannian 

manifold, Sci. Sinica Ser. A 27 (1984), no. 12, 1265–1273. MR 794292 (87a:58162)

RICHARD SCHOEN - SPECTRAL GEOMETRY (MATH 286 ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?