# The boundary value Minkowski problem for Weingarten curvatures

@article{Cruz2018TheBV, title={The boundary value Minkowski problem for Weingarten curvatures}, author={Fl{\'a}vio Cruz}, journal={arXiv: Differential Geometry}, year={2018} }

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.

## 17 References

### A GENERALIZED MINKOWSKI PROBLEM WITH DIRICHLET BOUNDARY CONDITION

- Mathematics, Philosophy
- 2000

. We prove the existence of hypersurfaces with prescribed boundary whose Weingarten curvature equals a given function that depends on the normal of the hypersurface.

### Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space

- Mathematics
- 2005

In this paper we consider the problem of finding a star-shaped
compact hypersurface with prescribed $k$-th mean curvature in
hyperbolic space. Under some sufficient conditions, we obtain an …

### Strictly Convex Submanifolds and Hypersurfaces of Positive Curvature

- Mathematics
- 2001

We construct smooth closed hypersurfaces of positive curvature with prescribed submanifolds and tangent planes. Further, we develop some applications to boundary value problems via Monge-Amp ere…

### Radial graphs of constant curvature and prescribed boundary

- Mathematics
- 2017

In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, without assuming the convexity of the prescribed…

### Convex hypersurfaces of prescribed curvature

- Mathematics
- 2002

For a smooth strictly convex closed hypersurface Σ in R, the Gauss map n : Σ → S is a diffeomorphism. A fundamental question in classical differential geometry concerns how much one can recover…

### CONSTANT HIGHER-ORDER MEAN CURVATURE HYPERSURFACES IN RIEMANNIAN SPACES

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2006

It is still an open question whether a compact embedded hypersurface in the Euclidean space $\mathbb{R}^{n+1}$ with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or…

### Degree theory for second order nonlinear elliptic operators and its applications

- Mathematics
- 1989

We introduce, along the lines of [4], an integer valued degree for second order fully nonlinear elliptic operators which is invariant under homotopy within elliptic operators. We also give some…

### The boundary value Minkowski problem. The parametric case

- Mathematics
- 1982

© Scuola Normale Superiore, Pisa, 1982, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze »…

### The Dirichlet problem for nonlinear second-order elliptic equations I

- Mathematics
- 1984

A green tire carrier having a wheeled frame providing support at three levels thereof for tire carrying sling members, each sling member being adjustable between several operative positions and an…

### Extrinsic geometry of convex surfaces

- Education
- 1973

If you really want to be smarter, reading can be one of the lots ways to evoke and realize. Many people who like reading will have more knowledge and experiences. Reading can be a way to gain…