arXiv research

In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To such a solution, we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data an…

The paper solves a curvature problem on a ball's surface near constant values.

In this paper we find strictly locally convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…

We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used to solve the Plateau problem for immersions of constant (and prescribed) Gaussian…

Upper and lower bounds for Steklov eigenvalues of submanifolds with fixed boundary.

Study finds strictly convex surfaces with specific curvature and boundary in space forms.

Study proves existence and nonexistence for annular surfaces with specific curvature and boundary.

Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.

Study of prescribing scalar curvature and mean curvature on compact manifolds with boundary.

Proves existence of curved surfaces in hyperbolic space.

Introduces electrostatic manifolds with boundary for curvature problems.

On a manifold with boundary, we deform the metric conformally. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in the interior, provided that there exists a subsolution.

Study finds conditions for metrics on curved spaces.

The paper solves curvature prescription problems on balls and disks.

Kapustin and Witten associate a Hecke modification of a holomorphic bundle over a Riemann surface to a singular monopole on a Riemannian surface times an interval satisfying prescribed boundary conditions. We prove existence and uniqueness of singular monopoles satisfying prescribed boundary conditions for any given He…

Proves existence of hypersurfaces with specific curvature properties.

In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, using the theory of fully nonlinear elliptic equations. We prove that if the given data admits a suitable radial graph as a subsolution, then we can find a radial graph with con…

Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.

Proves existence of convex surfaces with specific curvature in hyperbolic space.

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.

This paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.

Optimally regularizes boundaries in the Heisenberg group with prescribed curvature.

We consider a free boundary problem for the Willmore functional. Given a smooth domain $Ω$ in ${\mathbb R}^3$, we construct Willmore disks wich are critical in the class of surfaces meeting $\partial Ω$ orthogonally along their boundary and having small prescribed area. Using rescaling we first obtain constrained solut…

Proves existence of static vacuum metrics with specific boundary data.

These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Ampère and linearized Monge-Ampère equations. The prescribed affine mean curvature equation is a fully nonlinear, fourth order, geometri…

Paper finds solutions for convex surfaces in space forms with specific curvature.

Paper finds unique solutions for curved surfaces with specific gradient.

Study finds solutions to curvature equation with boundary conditions.

Compact capillary hypersurfaces with specific curvatures are proven.

The paper solves fractional combinatorial flows for prescribed hyperbolic bordered surfaces.

The article classifies curvature functions on compact manifolds with boundaries.

The paper finds convex hypersurfaces with specific curvature properties.

Study translating solitons in Minkowski space with prescribed Gauss image.

Solutions found for specific curvature problems in geometric settings.

Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.

Using the Perron method, we prove the existence of hypersurfaces of prescribed special Lagrangian curvature with prescribed boundary inside complete Riemannian manifolds of non-positive curvature.

Paper solves Dirichlet problem for $p$-convex hypersurfaces with curvature constraints.

The paper proves the existence and uniqueness of certain spacelike hypersurfaces with specific curvature and boundary conditions.

If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of a free boundary values variational problem. Such is, for instance, the problem of finding the shortest curve whose endpoints can slide along two prescribed …

Compactness results for hypersurfaces with mean curvature prescribed by an ambient function.

Study of metrics with prescribed curvature and geodesic curvature on a disc.

Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…

We study the prescribed mean curvature equation with a prescribed boundary contact angle condition in $M\times\R$ where $M^n$ is a Riemannian submanifold in $\R^{n+1}$. The main purpose is to establish a priori gradient estimates for solutions, from which the long time existence of the solution are derived.

The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.

We construct a sequence of compact embedded minimal disks in a ball in Euclidean 3-space, whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the x_3-axis. This extends a result of Colding and Minicozzi, who constructed a …

In this paper we prove classification results to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic ball on $\mathbb{S}^n$) of dimension $n\geq 2$ with prescribed constant mean curvat…

We consider smooth bounded surfaces with a smooth boundary and a prescribed background metric g_0. We now consider all metrics g conformal to g_0 which have a prescribed volume M. We now minimize the first eigenvalue of the Laplace operator of g over the metrics conformal to g_0 and having the prescribed volume. We sho…

We discuss the problem of prescribing the mean curvature and conformal class as boundary data for Einstein metrics on 3-manifolds, in the context of natural elliptic boundary value problems for Riemannian metrics.