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Let $C$ be a strictly convex domain in a $3$-dimensional Riemannian manifold with sectional curvature bounded above by a constant and let $\Sigma$ be a constant mean curvature surface with free boundary in $C$. We provide a pinching condition on the length of the traceless second fundamental form on $\Sigma$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if $C$ is a geodesic ball of $3$-dimensional space forms, then $\Sigma$ is either a spherical cap or a Delaunay surface., Comment: 23 pages

We derive linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball of a Riemannian manifold in terms of the modified volume. It is known that the twice of the area of a free boundary minimal surface in a Euclidean unit ball is equal to the length of its boundary. This can be extended to space forms by using our linear isoperimetric inequalities for the modified volume. Moreover, we obtain a sharp lower bound for the modified volume of free boundary minimal surfaces in a geodesic ball of the upper hemisphere $$\mathbb {S}_+^3$$ S + 3 . Finally, it is proved that the monotonicity property still holds for the modified volume of any submanifold in a complete simply connected Riemannian manifold with sectional curvature bounded above by a constant.

In this paper, we prove that a complete noncompact submanifold in a weighted manifold with nonpositive sectional curvature has at least linear weighted volume growth. Moreover we obtain several sufficient conditions for f -minimal hypersurfaces to have infinite weighted volume. By using an f -Laplacian comparison result, we obtain a lower bound of the first eigenvalue for the f -Laplace operator on submanifolds in a weighted manifold. We also obtain vanishing results for L f 2 harmonic 1-forms on complete noncompact f -minimal hypersurfaces in a weighted manifold. Finally we prove a topological structure theorem for complete noncompact L f -stable f -minimal hypersurfaces via a Liouville-type theorem for f -harmonic functions with finite f -energy.

Let N be a complete simply connected Riemannian manifold with sectional curvature K N satisfying − k 2 ≤ K N ≤ 0 for a nonzero constant k . In this paper we prove that if M is an n ( ≥ 3 ) -dimensional complete minimal hypersurface with finite index in N , then the space of L p harmonic 1-forms on M must be finite dimensional for certain p > 0 provided the bottom of the spectrum of the Laplace operator is sufficiently large. In particular, M has finitely many ends. These results can be regarded as an extension of Li–Wang (2002).

In this paper, we derive density estimates for submanifolds with variable mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. This leads to distance estimates for the boundaries of compact connected submanifolds. As applications, we give several necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small L 2 norm of the mean curvature vector in a Riemannian manifold.

In this paper, we estimate the p-fundamental tone of submanifolds in a Cartan–Hadamard manifold. First, we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension-one $$C^2$$ -foliations of open subsets $$\Omega $$ of Riemannian manifolds M and obtain lower bound estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of $$\Omega $$ .

In this paper, we study the connectedness at infinity of complete submanifolds by using the theory of p-harmonic function. For lower-dimensional cases, we prove that if M is a complete orientable noncompact hypersurface in $$\mathbb {R}^{n+1}$$ and if $$\delta $$ -stability inequality holds on M, then M has only one p-nonparabolic end. It is also proved that if $$M^n$$ is a complete noncompact submanifold in $${\mathbb {R}}^{n+k}$$ with sufficiently small $$L^n$$ norm of the traceless second fundamental form, then M has only one p-nonparabolic end. Moreover, we obtain a lower bound of the fundamental tone of the p Laplace operator on complete submanifolds in a Riemannian manifold.

Let M be an $$n(\ge $$ 3)-dimensional closed hypersurface in a unit sphere with constant m-th order mean curvature and with two distinct principal curvatures. We obtain a sharp curvature integral for M in terms of Ricci curvature, which gives a characterization of a Clifford hypersurface. Moreover we give a generalization of Simons’ integral inequality for closed hypersurface with vanishing m-th order mean curvature by making use of the Laplacian of the function of principal curvatures.

Let M be an n-dimensional complete orientable noncompact hypersurface in a complete Riemannian manifold of nonnegative sectional curvature. For 2 ≤ n ≤ 6 , we prove that if M satisfies the δ-stability inequality ( 0 δ ≤ 1 ), then there is no nontrivial L 2 β harmonic 1-form on M for some constant β. We also provide sufficient conditions for complete hypersurfaces to satisfy the δ-stability inequality. Moreover, we prove a vanishing theorem for L 2 harmonic 1-forms on M when M is an n-dimensional complete noncompact submanifold in a complete simply-connected Riemannian manifold N with sectional curvature K N satisfying that − k 2 ≤ K N ≤ 0 for some constant k.

Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ∩∂C and ∂Σ ∼ ∂C is radially connected from a point p ∈ ∂Σ∩∂C. We introduce a modified volume Mp(Σ) of Σ and obtain a sharp isoperimetric inequality where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume.

In this article we provide estimates on the first eigenvalue of stable minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded from below and above by negative constants. We also obtain a lower bound of the total scalar curvature of a stable minimal hypersurface if the scalar curvature of the ambient space is positive. Mathematics subject classification (2010): 53C21, 53A10.

For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz–Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilical point, which was developed by Choe [‘Sufficient conditions for constant mean curvature surfaces to be round’, Math. Ann.323(1) (2002), 143–156]. Using the concept of the rotation index at the interior and boundary umbilical points and applying the Poincaré–Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than four vertices in a domain of $\Bbb L^3$ bounded by (spacelike or timelike) totally umbilical surfaces is part of a (spacelike) plane or a hyperbolic plane. Moreover, we prove that the only immersed spacelike disk type capillary surface inside a de Sitter surface in $\Bbb L^3$ is part of (spacelike) plane or a hyperbolic plane.

In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature.

In this paper, we prove the nonexistence of $L^2$ harmonic 1-forms on a complete super stable minimal submanifold $M$ in hyperbolic space under the assumption that the first eigenvalue $\lambda_1 (M)$ for the Laplace operator on $M$ is bounded below by $(2n-1)(n-1)$. Moreover, we provide sufficient conditions for minimal submanifolds in hyperbolic space to be super stable., Comment: 6 pages, to appear in Journal of Mathematical Analysis and Applications

Let $M$ be an $n$-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-\kappa^2$. Using the cone total curvature $TC(\Gamma)$ of a graph $\Gamma$ which was introduced by Gulliver and Yamada Math. Z. 2006, we prove that the density at any point of a soap film-like surface $\Sigma$ spanning a graph $\Gamma \subset M$ is less than or equal to $\frac{1}{2\pi}\{TC(\Gamma) - \kappa^{2}\area(p\mbox{$\times\hspace*{-0.178cm}\times$}\Gamma)\}$. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when $n=3$, this density estimate implies that if \begin{eqnarray*} TC(\Gamma) < 3.649\pi + \kappa^2 \inf_{p\in M} \area({p\mbox{$\times\hspace*{-0.178cm}\times$}\Gamma}), \end{eqnarray*} then the only possible singularities of a piecewise smooth $(\mathbf{M},0,\delta)$-minimizing set $\Sigma$ is the $Y$-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $\pi/b$, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature., Comment: 14 pages

Let M n be a complete immersed super stable minimal sub-manifold in R n + p with °at normal bundle. We prove that if M has flnitetotal L 2 norm of its second fundamental form, then M is an a–ne n -plane. We also prove that any complete immersed super stable minimalsubmanifold with °at normal bundle has only one end. 1. IntroductionLet M be an n -dimensional complete minimal submanifold in R n + p . When n = 2 and p = 1, do Carmo and Peng [3], Fischer-Colbrie and Schoen [5]independently showed that the only complete stable minimal surface is a plane.Recall that a minimal submanifold is stable if the second variation of its volumeis always nonnegative for any normal variation with compact support. Forhigher dimensional minimal hypersurfaces, do Carmo and Peng [4] generalizedthe result mentioned as above. We will denote by A the second fundamentalform of M .Theorem ([4]). Let M n be a complete stable minimal hypersurface in R n +1 satisfying R M jAj 2 dv < 1. Then M must be a hyperplane.

Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a nonpositive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C. If ∂Σ ∼ ∂C is radially connected from a point \(p \in \partial\Sigma\cap \partial C\), then we prove a sharp relative isoperimetric inequality $$2\pi{\rm Area}(\Sigma) \leq {\rm Length}(\partial\Sigma \sim \partial C)^2 + K{\rm Area}(\Sigma)^2,$$ where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove the relative isoperimetric inequalities for minimal submanifolds outside a closed convex set in a higher-dimensional Riemannian manifold.

The classical isoperimetric inequality for a domain \(\Sigma \subset \mathbb{R}^{k}\) with smooth boundary \(\partial \Sigma\) is \(\displaystyle{ k^{k}\omega _{ k}\mathrm{Vol}(\Sigma )^{k-1} \leq \mathrm{ Vol}(\partial \Sigma )^{k}, }\) where equality holds if and only if \(\Sigma\) is a ball in \(\mathbb{R}^{k}\).

Let $\Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar\'{e} ball model $B^n$ of hyperbolic geometry. If we consider $\Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $\Sigma$ and the ideal boundary $\partial_\infty \Sigma$, say $\rvol(\Sigma)$ and $\rvol(\partial_\infty \Sigma)$, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if $\rvol(\partial_\infty \Sigma) \geq \rvol(\mathbb{S}^{k-1})$, then $\Sigma$ satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such $\Sigma$, we further obtain a sharp lower bound for the Euclidean volume $\rvol(\Sigma)$, which is an extension of Fraser and Schoen's recent result \cite{FS} to hyperbolic space. Moreover we introduce the M\"{o}bius volume of $\Sigma$ in $B^n$ to prove an isoperimetric inequality via the M\"{o}bius volume for $\Sigma$., Comment: 17 pages, 1 figure

We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface $M$ has sufficiently small total scalar curvature then $M$ has only one end. We also obtain a vanishing theorem for $L^2$ harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable., Comment: 16 pages

In this paper, we prove that minimal hypersurfaces when $n\geq 3$ and nonzero constant mean curvature hypersurfaces when $n\geq2$ foliated by spheres in parallel horizontal hyperplanes in ${\mathbb{H}}^n \times \mathbb{R}$ must be rotationally symmetric., This article is to appear in Bull. Korean Math. Soc. in 2010

We prove that if an $n$-dimensional complete minimal submanifold $M$ in hyperbolic space has sufficiently small total scalar curvature then $M$ has only one end. We also prove that for such $M$ there exist no nontrivial $L^2$ harmonic 1-forms on $M$.

In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface $M$ in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on $M$. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space., Comment: This article is to appear in J. Korean Math. Soc. in 2010

Let M be an n-dimensional complete minimal submanifold in Rn+p. Lei Ni proved that if M has sufficiently small total scalar curvature, then M has only one end. We improve the upper bound of total scalar curvature. We also prove that if M has the same upper bound of total scalar curvature, there is no nontrivial L2 harmonic 1-form on M.

Let $C$ be a closed convex set on a complete simply connected surface $S$ whose Gaussian curvature is bounded above by a nonpositive constant $K$. For a relatively compact subset $\Omega \subset S \sim C$, we obtain the sharp relative isoperimeric inequality $2\pi \mathrm{Area}(\Omega )-K\mathrm{Area}(\Omega )^{2} \leq \mathrm{Length}(\partial \Omega \sim \partial C)^{2}$. And we also have a similar partial result with positive Gaussian curvature bound.

We introduce a general homogeneous kernel whose degree is given by two parameters to establish the equivalent inequalities with the norm of a new Hilbert-type operator. As applications, we provide new extended Hilbert-type inequalities with the best possible constant factors.