Uniformization of surfaces with boundary and the application to the triple junction surfaces with negative Euler characteristic

Sat, 09 Oct 2021 11:51:14 +0800

Main results

The paper first develops a quantitative uniformization theorem for surfaces with boundary and then uses it to match the three sheets of a triple-junction surface.

Surfaces with boundary

Prescribed constant interior and boundary curvature

For a compact oriented surface with negative Euler characteristic and $(k,c)$ satisfying $k\leq0$ and $c<\sqrt{-k}$, there is a unique conformal metric with Gaussian curvature $k$ and constant boundary geodesic curvature $c$.

Quantitative control

Area and boundary length vary monotonically

The area $A(k,c)$ and boundary length $L(k,c)$ are continuous and satisfy sharp monotonicity and limiting laws, including the behavior of $cL(k,c)$ needed to balance junction curvature.

Triple junctions

A weak hyperbolic uniformization

If a triple-junction surface has connected junction and $\chi(M)\leq0$, its three sheets admit conformal hyperbolic metrics with a common boundary length whose geodesic curvatures sum to zero; the sheet metric is unique when its Euler characteristic is nonpositive.

Proof strategy

The prescribed-curvature problem is written as a semilinear elliptic boundary value problem. Maximum-principle estimates give existence, uniqueness, continuity, and monotonicity as $k$ and $c$ vary. For the junction problem, the boundary length is used as a common parameter: the three curvature-length functions are added, and their monotonicity and endpoint limits produce the unique balancing length.

Positive-curvature phenomenon

The following is a short video shows that why the uniqueness might fail when $K>0$.

In this video, we suppose the surface $\Sigma$ is the two copies of the domain enclosed by six arcs by identifying their corresponding edges marked with green color. So here $\Sigma$ is just a genus zero surface with three boundary components. We fix one boundary of $\Sigma$ having geodesic curvature $-1$ (marked as red) and the remaining two boundaries being geodesic (marked as black). Then we can let the area grow to see how the Gauss curvature changes after uniformization. The blue circle in this video represents the infinity circle if $K<0$ or the geodesic circle if $K>0$. From the video, we can see that $K$ will increase as area $A$ increases, then $K$ will reach its maximal point and after that $K$ will decrease to 0. So we will see that for some $K>0$, there are indeed two metrics with prescribed Gauss curvature and geodesic curvatures of boundaries.