arXiv research

Study on $t$-graphs with prescribed mean curvature in Heisenberg groups.

Study of prescribed mean curvature flow on noncompact hypersurfaces in Lorentz manifolds.

Proves optimal regularity for sphere minimizers in 3-sphere.

Theory proves existence of hypersurfaces with prescribed curvature.

We give a new existence proof for closed hypersurfaces of prescribed mean curvature in Lorentzian manifolds.

Constructs hyperspheres with prescribed mean curvature in Euclidean space.

It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds.

We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms.

The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is t…

Lectures on PDEs for creating surfaces with specific curvature.

We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclid…

We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski $3$-space.

It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.

We prove a semi-global result on the existence of conformal embeddings of the two-sphere into the round three-sphere S^3(1) with prescribed mean curvature.

Proves existence of graph on torus with prescribed curvature.

We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.

We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to $S^n$ for small $n, n\le6$, provided there are barriers.

Study phase transitions with prescribed mean curvature in Riemannian manifolds.

The paper finds closed hypersurfaces with prescribed mean curvature in non-compact manifolds.

Paper proves existence of PMC hypersurfaces in conformal product manifolds.

Study on curvature equation in Heisenberg group with convex boundary.

We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersur…

We consider two-dimensional immersions of disc-type in R^n. We focus well known classical concepts and study the nonlinear elliptic systems of such mappings. Using an Osserman-type condition we give a priori-estimates of the principle curvatures for certain graphs in R^4 with prescribed mean curvature.

We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a …

The paper proves the existence of H-spheres with arbitrary codimensions in certain Riemannian manifolds.

Classification of surfaces with prescribed mean curvature in Heisenberg space and SL2(R).

Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we study a conformally invariant spinor field equation on the $m$-sphere, $m\geq2$. Via variational methods and the spinorial Weierstraß representation, we study the problem of prescribing mean curvature for the imme…

The paper studies how certain surfaces evolve in space-time.

Survey on rigidity results for graphs with prescribed mean curvature.

In this paper we consider surfaces of class $C^1$ with continuous prescribed mean curvature in a three-dimensional contact sub-Riemannian manifold and prove that their characteristic curves are of class $C^2$. This regularity result also holds for critical points of the sub-Riemannian perimeter under a volume constrain…

Two spheres found with specific curvature constraints.

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in $\mathbb R^{n+1}$ over general domains $Ω$ without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result …

In this paper we consider a set $E\subsetΩ$ with prescribed mean curvature $f\in C(Ω)$ and Euclidean Lipschitz boundary $\partial E=Σ$ inside a three-dimensional contact sub-Riemannian manifold $M$. We prove that if $Σ$ is locally a regular intrinsic graph, the characteristic curves are of class $C^2$. The result is sh…

Study uses blow-up method to solve PMC problems with fixed boundaries.

We consider graphs Sigma^n in R^m with prescribed mean curvature and flat normal bundle. Using techniques of Schoen, Simon and Yau, and Ecker-Huisken, we derive an interior curvature estimate of the form |A|^2<=C/R^2 up to dimension n<=5, where C is a constant depending on natural geometric data of Sigma^n only. This g…

Develops a duality for graphs in Riemannian and Lorentzian spaces with prescribed mean curvature.

We prove the existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature in Cartan-Hadamard manifolds $N$. More precisely, given a suitable subset $L$ of the asymptotic boundary of $N$ and a suitable function $H$ on $N$, we are able to construct a set of locally finite perime…

We extend the interior gradient estimate due to N. Korevaar and L. Simon for solutions of the mean curvature equation from the case of Euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary …

Proves existence of sphere foliations with prescribed mean curvature on Riemannian manifolds.

In this paper a convergent series expansion is constructed to solve the prescribed mean curvature equation for n-dimensional hypersurfaces in n+1 dimensional Euclidean or Minkowskian space(time) which are graphs of a smooth real function u, and whose mean curvature function H is not too large in Hoelder norm, and integ…

The purpose of this paper is to study immersed surfaces in the product spaces $\mathbb{M}^2(κ)\times\mathbb{R}$, whose mean curvature is given as a $C^1$ function depending on their angle function. This class of surfaces extends widely, among others, the well-known theory of surfaces with constant mean curvature. In th…

We prove that any piece of a rotational hypersurface with prescribed mean curvature function in a Euclidean space can be uniquely extended infinitely, which generalizes the results by Euler and Delaunay for surfaces of revolution with constant mean curvautre. Next, we prove the same kind of theorem for generalized rota…

The paper constructs surfaces with prescribed mean curvature in a specific space.

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…

Study quantifies properties of PMC hypersurfaces with area bounds.

We present three ways to establish general stability inequalities for various classes of 2-immersions in Euclidean spaces of higher codimension

The paper studies weak singular Hermite-Einstein structures on homogeneous vector bundles.

The study proves manifold properties related to positive scalar curvature.