Counterexamples to Calabi conjectures on minimal hypersurfaces ...

3 The proofsLet ϕ: M m → N n be an isometric immersion between Riemannian manifolds. Givena function g ∈ C ∞ (N) we set f = g ◦ ϕ ∈ C ∞ (M). Sincefor every vec<strong>to</strong>r field X ∈ TM, we obtain〈grad M f, X〉 = 〈grad N g, X〉grad N g = grad M f + (grad N g) ⊥according <strong>to</strong> the decomposition TN = TM ⊕T ⊥ M. An easy computation using theGauss formula gives the well-known relation (see e.g. [8])Hess M f(X, Y ) = Hess N g(X, Y ) + 〈grad N g, α(X, Y )〉 (3)for all vec<strong>to</strong>r fields X, Y ∈ TM, where α stands for the second fundamental form ofϕ. In particular, taking traces with respect <strong>to</strong> an orthonormal frame {e 1 , . . .,e m }in TM yieldsm∑∆ M f = Hess N g(e i , e i ) + 〈grad N g, H〉. → (4)where → H= ∑ mi=1 α(e i, e i ).i=1The first main ingredient of our proofs is the Hessian comparison theorem.Theorem 3.1 Let M m be a Riemannian manifold and x 0 , x 1 ∈ M be such that thereis a minimizing unit speed geodesic γ joining x 0 and x 1 and let ρ(x) = dist(x 0 , x)be the distance function <strong>to</strong> x 0 . Let K γ ≤ b be the radial sectional curvatures of Malong γ. If b > 0 assume ρ(x 1 ) < π/2 √ b. Then, we have Hess ρ(x)(γ ′ , γ ′ ) = 0 andwhere X ∈ T x M is perpendicular <strong>to</strong> γ ′ (ρ(x)).Hess ρ(x)(X, X) ≥ C b (ρ(x))‖X‖ 2 (5)The second main ingredient is the version proved by Pigola-Rigoli-Setti [14,Theorem 1.9] of the Omori-Yau maximum principle.Theorem 3.2 Let M m be a Riemannian manifold and assume that there exists anon-negative C 2 -function ψ satisfying the following requirements:ψ(x) → +∞as x → ∞∃ A > 0 such that |gradψ| ≤ A √ ψ off a compact set√∃ B > 0 such that ∆ψ ≤ B ψG( √ ψ) off a compact set4