arXiv research

Study f-minimal hypersurfaces in weighted manifolds, finding index bounds.

We derive the Simons' type equation for $f$-minimal hypersurfaces in weighted Riemannian manifolds and apply it to obtain a pinching theorem for closed $f$-minimal hypersurfaces immersed in the product manifold $\mathbb{S}^n(\sqrt{2(n-1)})\times \mathbb{R}$ with $f=\frac {t^2}{4}$. Also we classify closed $f$-minimal h…

Study bounds index of minimal hypersurfaces in curved spaces.

We study stability properties of $f$-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li-Tam theory, we investigate the top…

Study on the index and Betti number of f-minimal hypersurfaces and self-shrinkers.

Let $(M^{n+1},g,e^{-f}dμ)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted vo…

In this paper, we study complete oriented $f$-minimal hypersurfaces properly immersed in a cylinder shrinking soliton $(\mathbb{S}^n\times \mathbb{R}, \bar{g}, f)$. We prove that such hypersurface with $L_f$-index one must be either $\mathbb{S}^n\times\{0\}$ or $\mathbb{S}^{n-1}\times\mathbb{R}$, where $\mathbb{S}^{n-1…

We study the asymptotic Dirichlet problem for $f$-minimal graphs in Cartan-Hadamard manifolds $M$. $f$-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of $f$-minimal graphs with pre…

Let $(M,\bar{g}, e^{-f}dμ)$ be a complete metric measure space with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove that, in $M$, there is no complete two-sided $L_f$-stable immersed $f$-minimal hypersurface with finite weighted volume. Further, if $M$ is a 3-manifold, we prove a smooth compa…

Let $Ω$ be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted Laplacian for closed embedded $f$-minimal hypersurfaces contained in $Ω$. Using this estimat…

In this paper we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of f-minimal hypersurfaces.

Eigenvalue estimates for weighted manifolds with applications.

Estimates the first eigenvalue of a Laplacian on self-shrinkers in Ricci shrinkers.

Study $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials.

New minimal tori found in curved spaces.

The paper extends Hamiltonian stability and mean curvature flow to Fano manifolds.

We classify (spacelike or timelike) surfaces of revolution with zero $f$-mean curvature in $\Bbb G^2\times\Bbb R_1,$ the Lorentz-Minkowski 3-space $\Bbb R^3_1$ endowed with the Gaussian-Euclidean density $e^{-f(x,y,z)}=\frac 1{2π}e^{-\frac{x^2+y^2}2}.$ It is proved that an $f$-maximal surface of revolution is either a …

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-Émery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold with positive $m…

The paper proves cohomology vanishing for a specific type of minimal submanifolds in a weighted Euclidean ball.

The paper studies stability and minimizing properties of higher codimensional surfaces in Euclidean space.

Study on lightlike hypersurfaces in metallic semi-Riemannian manifolds.

The paper classifies Codazzi hypersurfaces and characterizes minimal hypersurfaces in Nil^4.

Classification of hypersurfaces in homogeneous spaces with specific properties.

The paper studies special null hypersurfaces in spacetimes.

Study on biconservative hypersurfaces with constant scalar curvature in space forms.

The paper classifies various types of hypersurfaces in a product space.

The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.

Classifies hypersurfaces with constant isotropic curvature in space forms.

We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…

New global section found for geodesic flows on convex hypersurfaces.

New tensors capture intrinsic embedding data of conformal hypersurfaces.

The study characterizes canal hypersurfaces in Euclidean spaces and their curvature properties.

Survey on Dupin hypersurfaces and isoparametric hypersurfaces in spheres.

In this paper we introduce radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric. The study of these hypersurfaces is motivated by the fact that for indefinite almost Hermitian manifolds this class of lightlike hypersurfaces does not exist. We also establish that radical transversal…

Study on Dirac operators on lightlike hypersurfaces in 4D Lorentzian manifolds.

In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat L…

New types of null hypersurfaces found in Sasakian space-forms.

The paper classifies and studies conformally flat hypersurfaces in 4D space.

New compact mean convex hypersurfaces found for positive λ.

In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in $\mathbb E^4$. First, we deal with $δ(2)$-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of G…

The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.

Study on minimal hypersurfaces in a special normed space.

Mean curvature flow by parallel hypersurfaces is solved for isoparametric hypersurfaces.

Proves convexity of certain hypersurfaces with negative λ.

Classifies and describes hypersurfaces in Siklos spacetimes.

Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.

Paper characterizes a special hypersurface in 5D sphere.

Minimal hypersurfaces are the only $H$-tensional in 4D space forms.