Luciano Mari

Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds

Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable solution to the semilinear equation − Δu = f(u) on a Riemannian manifold with non-negative Ricci curvature, we are able to classify both the solution and the manifold. We also discuss the classification of monotone (with respect to the direction of some Killing vector field) solutions, in the spirit of a conjecture of De Giorgi, and the rigidity features for overdetermined elliptic problems on submanifolds with boundary.

Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

Abstract In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold ( M , 〈 , 〉 ) , namely the existence of a conformal deformation of the metric 〈 , 〉 realizing a given function s ˜ ( x ) as its scalar curvature. In particular, the work focuses on the case when s ˜ ( x ) changes sign. Our main achievement are two new existence results requiring minimal assumptions on the underlying manifold, and ensuring a control on the stretching factor of the conformal deformation in such a way that the conformally deformed metric be bi-Lipschitz equivalent to the original one. The topological–geometrical requirements we need are all encoded in the spectral properties of the standard and conformal Laplacians of M. Our techniques can be extended to investigate the existence of entire positive solutions of quasilinear equations of the type Δ p u + a ( x ) u p − 1 − b ( x ) u σ = 0 where Δ p is the p-Laplacian, σ > p − 1 > 0 , a , b ∈ L loc ∞ ( M ) and b changes sign, and in the process of collecting the material for the proof of our theorems, we have the opportunity to give some new insight on the subcriticality theory for the Schrodinger type operator Q V ′ : φ ⟼ − Δ p φ − a ( x ) | φ | p − 2 φ . In particular, we prove sharp Hardy-type inequalities in some geometrically relevant cases, notably for minimal submanifolds of the hyperbolic space.

Eigenvalue estimates for submanifolds of warped product spaces

We give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type

Erratum to “Some geometric properties of hypersurfaces with constant

N^n \times_f Q^q

r

, where

-mean curvature in Euclidean space”

f \in C^\infty(N)

Yamabe type equations with sign-changing nonlinearities on non-compact Riemannian manifolds

. We also study the essential spectrum of these minimal submanifolds.

Quasilinear Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem on non-compact manifolds

An erratum to the paper [D. Impera, L. Mari, and M. Rigoli, Some geometric properties of hypersurfaces with constant r-mean curvature in Euclidean space, Proc. Amer. Math. Soc. 139 (2011), no. 6, 2207-2215] is presented. In computation (13) of our previous paper [3] there are some inaccuracies concerning the constants. The computation should read as follows: S1Sj+1 − (j + 2)Sj+2 = m ( m j + 1 ) H1Hj+1 − (j + 2) ( m j + 2 ) Hj+2 = ( m j + 1 ) (mH1Hj+1 − (m− j − 1)Hj+2) ≥ ( m j + 1 ) (j + 1)H1Hj+1 ≥ 0, where the inequality is a consequence of (12). Hence, A(r)vj(r) ≥ (j + 1) ( m j + 1 ) Hj+1 ∫ ∂Br H1 = ( m−2 j ) Hj+1 m− j − 1 v1(r). Condition (2) of Theorem 1.1 should then read as follows: (ii) vj(r) −1 ∈ L(+∞) and lim inf r→+∞ √ v1(r)vj(r) ∫ +∞ r ds vj(s) > 1 2 [( m−2 j ) Hj+1 m− j − 1 ]−1/2 . Moreover, Remark 1.5 should be restated in this way: As we will see later, condition Sj+1 ≡ 0 together with rank(A) > j at every point of M implies the ellipticity of the operator Lj . Moreover, if we assume the additional hypothesis that there exists p ∈ M such that Hi(p) > 0 for every 1 ≤ i ≤ j, it can be proved that each Pi is positive definite for every 1 ≤ i ≤ j. Received by the editors October 6, 2011. 2010 Mathematics Subject Classification. Primary 53C21, 53C42; Secondary 58J50, 53A10. c ©2013 American Mathematical Society Reverts to public domain 28 years from publication 2221 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2222 DEBORA IMPERA, LUCIANO MARI, AND MARCO RIGOLI Similarly, Proposition 2.2(ii) has to be replaced by: (ii) Sj+1 ≡ 0, rank(A) > j at every point of M , and there exists p ∈ M such that Hi(p) > 0 for every 1 ≤ i ≤ j. The next remark should be added after Proposition 2.2. Remark. We stress that, by [10], when Sj+1 ≡ 0, the sole condition rank(A) > j is equivalent to the requirement that Lj be elliptic. Taking into account the previous observations, Theorem 1.4 has to be replaced by the following: Theorem 1.4. Let f : M → R be a complete, connected orientable hypersurface with Hj+1 ≡ 0, for some j ∈ {0, . . . ,m− 2}. If j ≥ 1, assume that rank(A) > j at every point. Furthermore, if j is even, suppose that there exists p ∈ M such that Hj(p) > 0. Set vj(r) = (m− j) ∫ ∂Bj |Sj |, vj+2(r) = ∫