Sharp Area Bounds for Free Boundary Minimal Surfaces in Conformally Euclidean Balls

Sharp Eigenvalue Bounds and Minimal Surfaces in the Ball

We prove existence and regularity of metrics on a surface with boundary which maximize σ1L where σ1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball Bn for some n. In the case of the annulus we prove that the unique solution to this problem is the induced metric on the cri...

Minimal Surfaces in Euclidean Space

Compact Embedded Minimal Surfaces of Positive Genus without Area Bounds

LetM be a three-manifold (possibly with boundary). We will show that, for any positive integer γ, there exists an open nonempty set of metrics on M (in the C-topology on the space of metrics on M) for each of which there are compact embedded stable minimal surfaces of genus γ with arbitrarily large area. This extends a result of Colding and Minicozzi, who proved the case γ = 1.

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons’ cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin’s theorem [43] is optimal.

Sharp Bounds for Eigenvalues and Multiplicities on Surfaces of Revolution

For surfaces of revolution diffeomorphic to S, it is proved that (S, can) provides sharp upper bounds for the multiplicities of all of the distinct eigenvalues. We also find sharp upper bounds for all the distinct eigenvalues and show that an infinite sequence of these eigenvalues are bounded above by those of (S, can). An example of such bounds for a metric with some negative curvature is pres...