Gregório Silva Neto - Academia.edu (original) (raw)

Papers by Gregório Silva Neto

Annali di Matematica Pura ed Applicata (1923 -)

In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean cu... more In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. These results were generalized by S. S. Chern, and then by Eschenburg and Tribuzy, for surfaces, homeomorphic to the sphere, in Riemannian manifolds with constant sectional curvature whose mean curvature function satisfies some bound on its differential. In this paper, using techniques partial differential equations in the complex plane which generalizes the notion of holomorphy, we extend these results for surfaces in a wide class of warped product manifolds, which includes, besides the classical space forms of constant sectional curvature, the de Sitter-Schwarzschild manifolds and the Reissner-Nordstrom manifolds, which are time slices of solutions of the Einstein field equations of the general relativity. Hilário Alencar and Gregório Silva Neto were partially supported by the National Council for Scientific and Technological Development-CNPq of Brazil.

Arkiv för Matematik, Oct 1, 2016

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional... more In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.

arXiv (Cornell University), Dec 13, 2020

In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder... more In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder as the only self-shrinkers satisfying a classic geometric assumption, namely the union of all tangent affine submanifolds of a complete self-shrinker omits a non-empty set of the Euclidean space. This assumption lead us to a new class of submanifolds, different from those with polynomial volume growth or the proper ones. We also prove an analogous result for self-expanders.

Journal of Differential Equations, Oct 1, 2023

In this paper, we deal with general divergence formulas involving symmetric endomorphisms. Using ... more In this paper, we deal with general divergence formulas involving symmetric endomorphisms. Using mild constraints in the sectional curvature and such divergence formulas we deduce a very general Poincaré type inequality. We apply such general inequality for higherorder mean curvature, in space forms and Einstein manifolds, to obtain several isoperimetric inequalities, as well as rigidity results for complete r-minimal hypersurfaces satisfying a suitable decay of the second fundamental form at infinity. Furthermore, using these techniques, we prove the flatness and non existence results for self-similar solutions of a large class of fully nonlinear curvature flows.

Mathematische Zeitschrift, Dec 27, 2013

We will prove that there are no stable complete hypersurfaces of R 4 with zero scalar curvature, ... more We will prove that there are no stable complete hypersurfaces of R 4 with zero scalar curvature, polynomial volume growth and such that (−K) H 3 ≥ c > 0 everywhere, for some constant c > 0, where K denotes the Gauss-Kronecker curvature and H denotes the mean curvature of the immersion. Our second result is the Bernstein type one there is no entire graphs of R 4 with zero scalar curvature such that (−K) H 3 ≥ c > 0 everywhere. At last, it will be proved that, if there exists a stable hypersurface with zero scalar curvature and (−K) H 3 ≥ c > 0 everywhere, that is, with volume growth greater than polynomial, then its tubular neighborhood is not embedded for suitable radius.

Journal of Differential Equations

In this paper, we deal with general divergence formulas involving symmetric endomorphisms. Using ... more In this paper, we deal with general divergence formulas involving symmetric endomorphisms. Using mild constraints in the sectional curvature and such divergence formulas we deduce a very general Poincaré type inequality. We apply such general inequality for higherorder mean curvature, in space forms and Einstein manifolds, to obtain several isoperimetric inequalities, as well as rigidity results for complete r-minimal hypersurfaces satisfying a suitable decay of the second fundamental form at infinity. Furthermore, using these techniques, we prove the flatness and non existence results for self-similar solutions of a large class of fully nonlinear curvature flows.

Results in Mathematics

In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder... more In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder as the only self-shrinkers satisfying a new geometric assumption, namely the union of all tangent affine submanifolds of a complete self-shrinker omits a non-empty set of the Euclidean space. This assumption lead us to a new class of submanifolds, different from those with polynomial volume growth or the proper ones (which was proved to be equivalent by Cheng and Zhou, see [5]). In fact, in the last section, we present an example of a non proper surface whose tangent planes omit the interior of a right circular cylinder, which proves that these classes are distinct from each other.

The Student Mathematical Library

arXiv (Cornell University), Mar 11, 2022

In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean cu... more In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. In this paper we survey some contributions of Renato Tribuzy to generalize Hopf's result as well as some recent results of the authors using these techniques for shrinking solitons of curvature flows and for surfaces in three-dimensional warped product manifolds, specially the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Hilário Alencar, Gregório Silva Neto and Detang Zhou were partially supported by the National Council for Scientific and Technological Development-CNPq of Brazil.

arXiv (Cornell University), Jun 16, 2021

This paper studies rigidity for immersed self-shrinkers of the mean curvature flow of surfaces in... more This paper studies rigidity for immersed self-shrinkers of the mean curvature flow of surfaces in the three-dimensional Euclidean space R 3. We prove that an immersed self-shrinker with finite L-index must be proper and of finite topology. As one of consequences, there is no stable two-dimensional self-shrinker in R 3 without assuming properness. We conclude the paper by giving an affirmative answer to a question of Mantegazza. Hilário Alencar, Gregório Silva Neto and Detang Zhou were partially supported by the National Council for Scientific and Technological Development-CNPq of Brazil.

(Impa). Atualmente é professor titular da Universidade Federal de Alagoas (Ufal), bolsista de pro... more (Impa). Atualmente é professor titular da Universidade Federal de Alagoas (Ufal), bolsista de produtividade do CNPq, editor da Coleção Profmat da Sociedade Brasileira de Matemática (SBM). Foi pró-reitor de pós-graduação e pesquisa da Ufal e presidente da SBM. Distinguido com a Ordem Nacional do Mérito Científico na Classe de Grã-Cruz e Associado Honorário da SBM. É membro titular da Academia Brasileira de Ciências e da Academia de Ciências do Mundo em Desenvolvimento (TWAS). Idealizou com Marcelo Viana o Profmat-Mestrado Profissional em Matemática em Rede Nacional. Walcy Santos Possui bacharelado em Matemática pela Universidade Federal do Rio de Janeiro (8), mestrado em Matemática pela Universidade Federal do Rio de Janeiro (8) e doutorado em Matemática pelo Instituto Nacional de Matemática Pura e Aplicada (). Atualmente é professora titular da Universidade Federal do Rio de Janeiro. Tem experiência na área de Matemática, com ênfase em Geometria e Topologia, atuando principalmente nos seguintes temas: geometria diferencial, r-curvatura média, formas espaciais, curvatura média e curvas planas. Foi Diretora do Instituto de Matemática da UFRJ de a 8.

arXiv (Cornell University), Jan 28, 2009

In 1968, Simons [9] introduced the concept of index for hypersurfaces immersed into the Euclidean... more In 1968, Simons [9] introduced the concept of index for hypersurfaces immersed into the Euclidean sphere S n+1. Intuitively, the index measures the number of independent directions in which a given hypersurface fails to minimize area. The earliest results regarding the index focused on the case of minimal hypersurfaces. Many such results established lower bounds for the index. More recently, however, mathematicians have generalized these results to hypersurfaces with constant mean curvature. In this paper, we consider hypersurfaces of constant mean curvature immersed into the sphere and give lower bounds for the index under new assumptions about the immersed manifold.

Annals of Global Analysis and Geometry, Mar 7, 2019

In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean cur... more In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary, oriented nonzero constant mean curvature surfaces in a large class of three dimensional warped product manifolds are embedded topological spheres, provided the mean curvature satisfies a positive lower bound depending only on the ambient curvatures. We conclude the paper proving that a stable, compact without boundary, nonzero constant mean curvature surface in a general Riemannian is a topological sphere provided its mean curvature has a lower bound depending only on the scalar curvature of the ambient space and the squared norm of the mean curvature vector field of the immersion of the ambient space in some Euclidean space. G. Silva Neto was partially supported by the National Council for Scientific and Technological Development-CNPq of Brazil.

Journal of Mathematical Analysis and Applications, Dec 1, 2023

Matemática Contemporânea, 2022

In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean cu... more In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. In this paper we survey some contributions of Renato Tribuzy to generalize Hopf's result as well as some recent results of the authors using these techniques for shrinking solitons of curvature flows and for surfaces in three-dimensional warped product manifolds, specially the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.

arXiv (Cornell University), Sep 22, 2020

In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily co... more In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in R 3. We show this self-similar solutions are the round spheres centered at the origin provided it has genus zero and satisfies a suitable upper pinching estimate for the Gaussian curvature. As applications, we obtain rigidity results for the round sphere as the only closed, immersed, genus zero, self-similar solution of several well known flows, as the flow of the powers of mean curvature, the harmonic mean curvature flow and the α-Gaussian curvature flow for α ∈ (0, 1/4). We remark that our result does not assume any embeddedness condition.

Asian Journal of Mathematics, 2017

In the first part of this paper we prove some new Poincaré inequalities, with explicit constants,... more In the first part of this paper we prove some new Poincaré inequalities, with explicit constants, for domains of any hypersurface of a Riemannian manifold with sectional curvatures bounded from above. This inequalities involve the first and the second symmetric functions of the eigenvalues of the second fundamental form of such hypersurface. We apply these inequalities to derive some isoperimetric inequalities and to estimate the volume of domains enclosed by compact self-shrinkers in terms of its scalar curvature. In the second part of the paper we prove some mean value inequalities and as consequences we derive some monotonicity results involving the integral of the mean curvature. ≤ C(p, Ω) ˆΩ |∇f | p dx 1 p. This is the Poincaré-Wirtinger inequality. An interesting question about these inequalities is to know the dependence of the Poincaré constant C(p, Ω) on the geometry of the domain Ω or H. Alencar was partially supported by CNPq of Brazil 2010 Mathematics Subject Classification. 53C21, 53C42.

Arkiv för Matematik, Apr 1, 2016

Revista Matematica Iberoamericana, Dec 6, 2018

In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Si... more In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Moreover, the equality is attained. Next, we prove some monotonicity formulas for submanifolds with bounded mean curvature vector in warped product manifolds and, as consequences, we give lower bound estimates for the volume of these submanifolds in terms of the warping function. We conclude the paper with an isoperimetric inequality for minimal surfaces.

Crelle's Journal, Oct 28, 2021

In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immerse... more In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three dimensional Euclidean space R 3 is a round sphere, provided its mean curvature and the norm of its position vector have an upper bound in terms of the norm of its traceless second fundamental form. The example constructed by Drugan justifies that the hypothesis on the second fundamental form is necessary. We can also prove the same kind of rigidity results for surfaces with parallel weighted mean curvature vector in R n with radial weight. These results are applications of a new generalization of Cauchy's Theorem in complex analysis which concludes that a complex function is identically zero or its zeroes are isolated if it satisfies some weak holomorphy.

Annali di Matematica Pura ed Applicata (1923 -)

In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean cu... more In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. These results were generalized by S. S. Chern, and then by Eschenburg and Tribuzy, for surfaces, homeomorphic to the sphere, in Riemannian manifolds with constant sectional curvature whose mean curvature function satisfies some bound on its differential. In this paper, using techniques partial differential equations in the complex plane which generalizes the notion of holomorphy, we extend these results for surfaces in a wide class of warped product manifolds, which includes, besides the classical space forms of constant sectional curvature, the de Sitter-Schwarzschild manifolds and the Reissner-Nordstrom manifolds, which are time slices of solutions of the Einstein field equations of the general relativity. Hilário Alencar and Gregório Silva Neto were partially supported by the National Council for Scientific and Technological Development-CNPq of Brazil.

Arkiv för Matematik, Oct 1, 2016

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional... more In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.

arXiv (Cornell University), Dec 13, 2020

In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder... more In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder as the only self-shrinkers satisfying a classic geometric assumption, namely the union of all tangent affine submanifolds of a complete self-shrinker omits a non-empty set of the Euclidean space. This assumption lead us to a new class of submanifolds, different from those with polynomial volume growth or the proper ones. We also prove an analogous result for self-expanders.

Journal of Differential Equations, Oct 1, 2023

In this paper, we deal with general divergence formulas involving symmetric endomorphisms. Using ... more In this paper, we deal with general divergence formulas involving symmetric endomorphisms. Using mild constraints in the sectional curvature and such divergence formulas we deduce a very general Poincaré type inequality. We apply such general inequality for higherorder mean curvature, in space forms and Einstein manifolds, to obtain several isoperimetric inequalities, as well as rigidity results for complete r-minimal hypersurfaces satisfying a suitable decay of the second fundamental form at infinity. Furthermore, using these techniques, we prove the flatness and non existence results for self-similar solutions of a large class of fully nonlinear curvature flows.

Mathematische Zeitschrift, Dec 27, 2013

We will prove that there are no stable complete hypersurfaces of R 4 with zero scalar curvature, ... more We will prove that there are no stable complete hypersurfaces of R 4 with zero scalar curvature, polynomial volume growth and such that (−K) H 3 ≥ c > 0 everywhere, for some constant c > 0, where K denotes the Gauss-Kronecker curvature and H denotes the mean curvature of the immersion. Our second result is the Bernstein type one there is no entire graphs of R 4 with zero scalar curvature such that (−K) H 3 ≥ c > 0 everywhere. At last, it will be proved that, if there exists a stable hypersurface with zero scalar curvature and (−K) H 3 ≥ c > 0 everywhere, that is, with volume growth greater than polynomial, then its tubular neighborhood is not embedded for suitable radius.

Journal of Differential Equations

In this paper, we deal with general divergence formulas involving symmetric endomorphisms. Using ... more In this paper, we deal with general divergence formulas involving symmetric endomorphisms. Using mild constraints in the sectional curvature and such divergence formulas we deduce a very general Poincaré type inequality. We apply such general inequality for higherorder mean curvature, in space forms and Einstein manifolds, to obtain several isoperimetric inequalities, as well as rigidity results for complete r-minimal hypersurfaces satisfying a suitable decay of the second fundamental form at infinity. Furthermore, using these techniques, we prove the flatness and non existence results for self-similar solutions of a large class of fully nonlinear curvature flows.

Results in Mathematics

In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder... more In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder as the only self-shrinkers satisfying a new geometric assumption, namely the union of all tangent affine submanifolds of a complete self-shrinker omits a non-empty set of the Euclidean space. This assumption lead us to a new class of submanifolds, different from those with polynomial volume growth or the proper ones (which was proved to be equivalent by Cheng and Zhou, see [5]). In fact, in the last section, we present an example of a non proper surface whose tangent planes omit the interior of a right circular cylinder, which proves that these classes are distinct from each other.

The Student Mathematical Library

arXiv (Cornell University), Mar 11, 2022

In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean cu... more In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. In this paper we survey some contributions of Renato Tribuzy to generalize Hopf's result as well as some recent results of the authors using these techniques for shrinking solitons of curvature flows and for surfaces in three-dimensional warped product manifolds, specially the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Hilário Alencar, Gregório Silva Neto and Detang Zhou were partially supported by the National Council for Scientific and Technological Development-CNPq of Brazil.

arXiv (Cornell University), Jun 16, 2021

This paper studies rigidity for immersed self-shrinkers of the mean curvature flow of surfaces in... more This paper studies rigidity for immersed self-shrinkers of the mean curvature flow of surfaces in the three-dimensional Euclidean space R 3. We prove that an immersed self-shrinker with finite L-index must be proper and of finite topology. As one of consequences, there is no stable two-dimensional self-shrinker in R 3 without assuming properness. We conclude the paper by giving an affirmative answer to a question of Mantegazza. Hilário Alencar, Gregório Silva Neto and Detang Zhou were partially supported by the National Council for Scientific and Technological Development-CNPq of Brazil.

(Impa). Atualmente é professor titular da Universidade Federal de Alagoas (Ufal), bolsista de pro... more (Impa). Atualmente é professor titular da Universidade Federal de Alagoas (Ufal), bolsista de produtividade do CNPq, editor da Coleção Profmat da Sociedade Brasileira de Matemática (SBM). Foi pró-reitor de pós-graduação e pesquisa da Ufal e presidente da SBM. Distinguido com a Ordem Nacional do Mérito Científico na Classe de Grã-Cruz e Associado Honorário da SBM. É membro titular da Academia Brasileira de Ciências e da Academia de Ciências do Mundo em Desenvolvimento (TWAS). Idealizou com Marcelo Viana o Profmat-Mestrado Profissional em Matemática em Rede Nacional. Walcy Santos Possui bacharelado em Matemática pela Universidade Federal do Rio de Janeiro (8), mestrado em Matemática pela Universidade Federal do Rio de Janeiro (8) e doutorado em Matemática pelo Instituto Nacional de Matemática Pura e Aplicada (). Atualmente é professora titular da Universidade Federal do Rio de Janeiro. Tem experiência na área de Matemática, com ênfase em Geometria e Topologia, atuando principalmente nos seguintes temas: geometria diferencial, r-curvatura média, formas espaciais, curvatura média e curvas planas. Foi Diretora do Instituto de Matemática da UFRJ de a 8.

arXiv (Cornell University), Jan 28, 2009

In 1968, Simons [9] introduced the concept of index for hypersurfaces immersed into the Euclidean... more In 1968, Simons [9] introduced the concept of index for hypersurfaces immersed into the Euclidean sphere S n+1. Intuitively, the index measures the number of independent directions in which a given hypersurface fails to minimize area. The earliest results regarding the index focused on the case of minimal hypersurfaces. Many such results established lower bounds for the index. More recently, however, mathematicians have generalized these results to hypersurfaces with constant mean curvature. In this paper, we consider hypersurfaces of constant mean curvature immersed into the sphere and give lower bounds for the index under new assumptions about the immersed manifold.

Annals of Global Analysis and Geometry, Mar 7, 2019

In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean cur... more In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary, oriented nonzero constant mean curvature surfaces in a large class of three dimensional warped product manifolds are embedded topological spheres, provided the mean curvature satisfies a positive lower bound depending only on the ambient curvatures. We conclude the paper proving that a stable, compact without boundary, nonzero constant mean curvature surface in a general Riemannian is a topological sphere provided its mean curvature has a lower bound depending only on the scalar curvature of the ambient space and the squared norm of the mean curvature vector field of the immersion of the ambient space in some Euclidean space. G. Silva Neto was partially supported by the National Council for Scientific and Technological Development-CNPq of Brazil.

Journal of Mathematical Analysis and Applications, Dec 1, 2023

Matemática Contemporânea, 2022

In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean cu... more In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. In this paper we survey some contributions of Renato Tribuzy to generalize Hopf's result as well as some recent results of the authors using these techniques for shrinking solitons of curvature flows and for surfaces in three-dimensional warped product manifolds, specially the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.

arXiv (Cornell University), Sep 22, 2020

In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily co... more In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in R 3. We show this self-similar solutions are the round spheres centered at the origin provided it has genus zero and satisfies a suitable upper pinching estimate for the Gaussian curvature. As applications, we obtain rigidity results for the round sphere as the only closed, immersed, genus zero, self-similar solution of several well known flows, as the flow of the powers of mean curvature, the harmonic mean curvature flow and the α-Gaussian curvature flow for α ∈ (0, 1/4). We remark that our result does not assume any embeddedness condition.

Asian Journal of Mathematics, 2017

In the first part of this paper we prove some new Poincaré inequalities, with explicit constants,... more In the first part of this paper we prove some new Poincaré inequalities, with explicit constants, for domains of any hypersurface of a Riemannian manifold with sectional curvatures bounded from above. This inequalities involve the first and the second symmetric functions of the eigenvalues of the second fundamental form of such hypersurface. We apply these inequalities to derive some isoperimetric inequalities and to estimate the volume of domains enclosed by compact self-shrinkers in terms of its scalar curvature. In the second part of the paper we prove some mean value inequalities and as consequences we derive some monotonicity results involving the integral of the mean curvature. ≤ C(p, Ω) ˆΩ |∇f | p dx 1 p. This is the Poincaré-Wirtinger inequality. An interesting question about these inequalities is to know the dependence of the Poincaré constant C(p, Ω) on the geometry of the domain Ω or H. Alencar was partially supported by CNPq of Brazil 2010 Mathematics Subject Classification. 53C21, 53C42.

Arkiv för Matematik, Apr 1, 2016

Revista Matematica Iberoamericana, Dec 6, 2018

In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Si... more In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Moreover, the equality is attained. Next, we prove some monotonicity formulas for submanifolds with bounded mean curvature vector in warped product manifolds and, as consequences, we give lower bound estimates for the volume of these submanifolds in terms of the warping function. We conclude the paper with an isoperimetric inequality for minimal surfaces.

Crelle's Journal, Oct 28, 2021

In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immerse... more In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three dimensional Euclidean space R 3 is a round sphere, provided its mean curvature and the norm of its position vector have an upper bound in terms of the norm of its traceless second fundamental form. The example constructed by Drugan justifies that the hypothesis on the second fundamental form is necessary. We can also prove the same kind of rigidity results for surfaces with parallel weighted mean curvature vector in R n with radial weight. These results are applications of a new generalization of Cauchy's Theorem in complex analysis which concludes that a complex function is identically zero or its zeroes are isolated if it satisfies some weak holomorphy.