相关论文: Free boundary minimal M\"obius band in spherical c…

In real space forms, Fraser and Schoen proved that a free-boundary minimal disk in a geodesic ball is totally geodesic. In this note, we consider free-boundary minimal surfaces $\Sigma$ (of any genus) in geodesic balls of complex space…

For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $\Sigma$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in…

We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already…

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of…

We prove that the area of each nonflat genus zero free boundary minimal surface embedded in the unit $3$-ball is less than the area of its radial projection to $\mathbb{S}^2$. The inequality is asymptotically sharp, and we prove any…

Let $(\Omega^{n+1}, g)$ be an $(n+1)$-dimensional smooth compact connected Riemannian manifold with smooth boundary $\Sigma$, satisfying that ${\text{Ric}_{\Omega}}\ge 0$ and $\Sigma$ is strictly convex, more precisely, its second…

We prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean $3$-ball in exactly two connected surfaces. We also show that if a region in the ball has mean convex boundary…

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the…

Let $M$ be a complete Riemannian $3$-manifold with sectional curvatures between $0$ and $1$. A minimal $2$-sphere immersed in $M$ has area at least $4\pi$. If an embedded minimal sphere has area $4\pi$, then $M$ is isometric to the unit…

We prove that a free boundary curve shortening flow on closed surfaces with a strictly convex boundary remains noncollapsed for a finite time in the sense of the reflected chord-arc profile introduced by Langford-Zhu. This shows that such…

This paper forms part of our ongoing works on the existence of complete non-compact free boundary minimal planes in an asymptotically flat three-dimensional Riemannian manifold with boundary. We set up the degree theory for the space of…

We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a…

This is the second of two articles in which we investigate the geometry of free boundary and capillary minimal surfaces in balls $B_R\subset\mathbb{S}^3$. In this article, we find monotonicity formulae which imply that capillary minimal…

We introduce a flow that is designed to flow maps $u:\Sigma\to \mathbb{R}^n$ which map the boundary of a general domain surface $\Sigma$ into a given (not necessarily connected) submanifold $N\hookrightarrow \mathbb{R}^n$ towards a free…

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold $\Omega$ with smooth boundary. We give a computable, sharp lower bound of the first eigenvalue of this problem, which depends only on the dimension, a lower…

We show several results comparing sharp eigenvalue bounds for the first Steklov eigenvalue on surfaces under change of the topology. Among others, we obtain strict monotonicity in the genus. Combined with results of the second named author…

We prove that compact 3-manifolds $M$ of constant curvature +1 with boundary a minimal surface are locally naturally parametrized by the conformal class of the boundary metric $\gamma$ in the Teichmuller space of $\partial M$, when…

In this paper, we consider locally wedge-shaped manifolds, which are Riemannian manifolds that are allowed to have both boundary and certain types of edges. We define and study the properties of free boundary minimal hypersurfaces inside…

In this work, we present several rigidity results for compact free boundary hypersurfaces in initial data sets with boundary. Specifically, in the first part of the paper, we extend the local splitting theorems from [G. J. Galloway and H.…

We employ min-max techniques to show that the unit ball in $\mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.