arXiv subjects

Flávio França Cruz

Publications and source records attributed to Flávio França Cruz.

3 recordsLinked to original sources

arXiv2025

An extension of Liebmann's Theorem to hypersurfaces with boundary

Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary. More precisely, we prove that a locally convex, embedded, compact, connected CMC hypersurface bounded by a closed strictly convex $(n-1)$-dimensional submanifold in a hyperplane $Π^n\subset \mathbb{R}^{n+1}$ lies in one of the two halfspace determined by $Π$ and inherits the symmetries of the boundary. Consequently, spherical caps are the only such hypersurfaces with non-zero constant mean curvature bounded by a $(n-1)-$sphere.

math.DG↗

arXiv2019

On the prescribed $Q$-curvature problem in Riemannian manifolds

We prove the existence of metrics with prescribed $Q$-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function on $\mathbb{R}^n$ can be realized as the $Q$-curvature of a Riemannian metric.

math.DG↗

arXiv2018

The boundary value Minkowski problem for Weingarten curvatures

In this paper, we prove the existence of hypersurfaces in the Euclidean space with prescribed boundary and whose k-th Weingarten curvature equals a given function that depends on the normal of the hypersurface. The proof is based on the solvability of a fully nonlinear elliptic PDE. The required a priori estimates are established under the natural assumptions that the prescribed boundary is strictly convex and the prescribed function satisfies a Serrin type condition.

math.DG↗