相关论文: A Characterization of the Critical Catenoid

In this paper we use the Alexandrov Reflection Method to obtain a characterization to embedded CMC capillary annulus $\Sigma^2 \subset \mathbb{B}^3$. In especial, but using a new strategy, we present a new characterization to the critical…

We develop new methods to compare the span $\mathcal{C}(\Sigma)$ of the coordinate functions on a free boundary minimal submanifold $\Sigma$ embedded in the unit $n$-ball $\mathbb{B}^n$ with its first Steklov eigenspace…

We prove that the only embedded free boundary minimal surface $\Sigma$ in $\mathbb{B}^3$ with index $4$ is the critical catenoid. This extends fundamental work of A. Fraser and R. Schoen, as well as the work of H. Tran.

We first consider a uniqueness problem for embedded free boundary minimal annuli in the three-dimensional Euclidean unit half-ball. Then, we obtain symmetry properties for compact embedded free boundary minimal surfaces in the unit ball.…

We consider $\Sigma$ an embedded free boundary minimal annulus in a geodesic ball in the round hemisphere $\mathbb{S}^3_+$ or in the hyperbolic space $\mathbb{H}^3$. Under the hypothesis of invariance due to an antipodal map on the geodesic…

We show that a minimal surface meeting a sphere at a 90-degree angle can be reflected across the sphere. Using this reflection, we prove the uniqueness that every embedded free boundary minimal annulus in a ball is necessarily the critical…

In this note we investigate free boundary minimal surfaces in the Euclidean 3-space, and by using holomorphic techniques developed by Fraser and Schoen we prove that the free boundary minimal annulus is the critical catenoid.

We construct a family of compact free boundary minimal annuli immersed in the unit ball $\mathbb{B}^3$ of $\mathbb{R}^3$, the first such examples other than the critical catenoid. This solves a problem formulated by Nitsche in 1985. These…

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue sigma_1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Sigma with genus gamma and k boundary…

We prove existence and regularity of metrics on a surface with boundary which maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free…

In this note, we use a result of Osserman and Schiffer \cite{OS} to give a variational characterization of the catenoid. Namely, we show that subsets of the catenoid minimize area within a geometrically natural class of minimal annuli. To…

In this article, we show that the critical catenoid, as a free boundary minimal surface of the unit ball in $\mathbb{R}^3$, has index $4$. We also prove that a free boundary minimal surface of the unit ball in $\mathbb{R}^3$, that is not a…

We prove a uniqueness result for free boundary minimal annuli in the unit Euclidean three-ball that are $\sigma$-homothetic to the critical catenoid.

In this paper we prove that the Morse index of the critical M\"obius band in the $4-$dimensional Euclidean ball $\mathbb B^4$ equals 5. It is conjectured that this is the only embedded non-orientable free boundary minimal surface of index 5…

We explicitly classify all $S^1$-invariant free boundary minimal annuli and M\"obius bands in $\mathbb{B}^n$. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $S^1$-invariant metrics on…

In 1970, Lawson solved the topological realization problem for minimal surfaces in the sphere, showing that any closed orientable surface can be minimally embedded in $\mathbb{S}^3$. The analogous problem for surfaces with boundary was…

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies…

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.

An embedded free boundary minimal surface in the 3-ball has a Steklov eigenvalue of one due to its coordinate functions. Fraser and Li conjectured that whether one is the first nonzero Steklov eigenvalue. In this paper, we show that if an…

For a compact, connected, orientable Riemannian manifold with $b$ boundary components, we obtain geometric lower bounds for the low Steklov eigenvalues, namely $\sigma_k$, $1\le k\le b-1$. Our results complement earlier results, which apply…