Symmetry of properly embedded special Weingarten surfaces in $\mathbf {H}^3$

Abstract

Abstract. In this paper we prove some existence and uniqueness results about special Weingarten surfaces in hyperbolic space.

Variants on Alexandrov reflection principle and other applications of maximum principle

“…An amazing fact is that Alexandrov Reflection yield at once several symmetry and uniqueness results about properly embedded constant mean curvature H surfaces in hyperbolic space: For instance, if the asymptotic boundary is a point or a circle then one gets an horosphere or an equidistant surface (see [29]), if the asymptotic boundary is the union of two disjoint circles and H = 0 (embeddedness hère is not necessary) then one gets a hyperbolic catenoid (see [62]), if the asymptotic boundary is the union of two disjoint circles and H * 0 then one gets a surface of révolution (see [30]). As a matter of fact, similar results hold for ƒ-surfaces in hyperbolic space as we have remarked in a recent paper (see [92]). Reeen tly, the authors have obtained some new symmetry results for constant mean curvature surfaces in hyperbolic space (see [92] and [93]).…”

“…As a matter of fact, similar results hold for ƒ-surfaces in hyperbolic space as we have remarked in a recent paper (see [92]). Reeen tly, the authors have obtained some new symmetry results for constant mean curvature surfaces in hyperbolic space (see [92] and [93]). The authors have also inferred some gênerai uniqueness (and existence) results for minimal vertical graphs in hyperbolic space (see [94]).…”

“…Next, we will extend a symmetry resuit deduced by the authors (see [92], Theorem land [93],Theorem4.2). THEOREM A.…”

“…Notice that the assumptions yield that M has non-zero mean curvature; this can be seen by applying Maximum Principle, using minimal hyperbolic catenoids as useful barriers. Of course, the above theorem is valid for ƒ-surfaces in hyperbolic space (see [92]).…”

“…Let M be a connected properly embedded constant mean curvature 1 surface in hyperbolic space contained inside the horosphere 36 and let dM be a circle <g lying on 3e. IfM is compact then it follows that M is part of an horosphere (see [72], |8], [92]). Thus we can assume that M is non-compact (no assumptions about the topology).…”