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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

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 $$\Omega $$ .

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.

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 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 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.

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.