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Marcio Batista

Publications and source records attributed to Marcio Batista.

7 recordsLinked to original sources
arXiv2024

First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces

In this paper, we obtain geometric upper bounds for the first eigenvalue $\lambda_1(J)$ of the Jacobi operator for both closed hypersurfaces and compact hypersurfaces with boundary having constant mean curvature (CMC). As an application, we derive new rigidity results for the area of CMC hypersurfaces under suitable conditions on $\lambda_1(J)$ and the curvature of the ambient space. We also address the Jacobi--Steklov problem, proving geometric upper bounds for its first eigenvalue $\sigma_1(J)$ and deriving rigidity results related to the length of the boundary. Additionally, we present some results in higher dimensions related to the Yamabe invariants.

math.DG↗
arXiv2018

Rigidity of manifolds admitting stable solutions of an elliptic problem

In this paper, we study geometric rigidity of Riemannian manifolds admitting stable solutions of certain elliptic problems (stability in a variational sense), that is, under suitable hypotheses, we are able to characterize the Riemannian manifold which admits a stable solution. Furthermore, under the non-negativity of the weighted Ricci curvature, we deduce several data about the stable solution and a splitting result for the manifold.

math.DG↗
arXiv2016

Hardy and Rellich Inequalities for submanifolds in Hadamard spaces

Some of the most known integral inequalities are the Sobolev, Hardy and Rellich inequalities in Euclidean spaces. In the context of submanifolds, the Sobolev inequality was proved by Michael-Simon and Hoffman-Spruck. Since then, a sort of applications to the submanifold theory has been derived from those inequalities. Years later, Carron obtained a Hardy inequality for submanifolds in Hadamard spaces. In this paper, we prove the general Hardy and Rellich Inequalities for submanifolds in Hadamard spaces. Some applications are given and we also analyse the equality cases.

math.DG↗
arXiv2014

Some Isoperimetric Inequalities and Eigenvalue Estimates in Weighted Manifolds

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.

math.DG↗
arXiv2012

Integral estimates for the trace of symmetric operators

Let $Φ:TM\to TM$ be a positive-semidefinite symmetric operator of class $C^1$ defined on a complete non-compact manifold $M$ isometrically immersed in a Hadamard space $\bar{M}$. In this paper, we given conditions on the operator $Φ$ and on the second fundamental form to guarantee that either $Φ\equiv 0$ or the integral $\int_M \mathrm{tr}\,ΦdM$ is infinite. We will given some applications. The first one says that if $M$ admits an integrable distribution whose integrals are minimal submanifolds in $\bar{M}$ then the volume of $M$ must be infinite. Another application states that if the sectional curvature of $\bar{M}$ satisfies $\bar{K}\leq -c^2$, for some $c\geq 0$, and $λ:M^m\to [0,\infty)$ is a nonnegative $C^1$ function such that gradient vector of $λ$ and the mean curvature vector $H$ of the immersion satisfy $|H+p\nabla λ|\leq (m-1)c λ$, for some $p\geq 1$, then either $λ\equiv 0$ or the integral $\int_M λ^s dM$ is infinite, for all $1\leq s\leq p$.

math.DG↗