arXiv research

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…

The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.

New method approximates Gaussian curvature on discrete surfaces.

New approach to prescribing Gaussian curvature on spheres with conical singularities.

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

We extend recent results of Guan and Spruck, proving existence results for constant Gaussian curvature hypersurfaces in Hadamard manifolds.

The paper solves curvature prescription on a disk with negative Gaussian curvature.

Study on compact Kähler surfaces for sign-changing curvatures.

Paper solves curvature assignment on surfaces with sharp points and edges.

The paper solves a curvature problem in hyperbolic space using a flow approach.

The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.

In this note, we prove that the abstract gradient flow introduced by Baird-Fardoun-Regbaoui \cite{BFR}is well-posed on a closed Riemann surface with conical singularity. Long time existence and convergence of the flow are proved under certain assumptions. As an application, the prescribed Gaussian curvature problem is …

The paper solves curvature prescription problems on balls and disks.

The paper finds circle packings with specific curvatures in hyperbolic geometry.

The abstract finds conditions for creating curves of constant curvature.

The paper solves a geometric problem using curvature flow and variational methods.

The paper solves a problem in metric geometry for disks with negative curvature.

Solves curvature prescription on rotational surfaces.

Let M be a compact Riemannian manifold and E a Riemannian vector bundle on M. We look for hypersurfaces of E with a prescribed vertical Gaussian curvature. In trying to solve this problem fibre-wise, we loose the regularity of the resulting solution. To unsure the smoothness of the solution, we construct it as a radial…

Paper finds conformal metrics on a disk with specific curvatures.

Prescribing $σ_k$ curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. Given a positive function $K$ to be prescribed on the 4-dimensional round sphere. We obtain asymptotic profile analysis for potentially blowing up solutions to the $σ_2$ curvature equatio…

We prove new results on existence of solutions for the prescribed gaussian curvature problem on the euclidean sphere S^2. Those results are achieved by relating this problem with the holomorphic triples theory on Riemann surfaces. We think this approach might be applied to study some other semi-linear elliptic equation…

New neural network solves Nirenberg problem for curvature on sphere.

The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(Σ,β)$ be a closed Riemann surface with a divisor $β$, and $K_λ=K+λ$, where $K:Σ\rightarrow\mathbb{R}$ is a Hölder continuous function satisfying $\max_ΣK= 0$, $K\not\equiv 0$, and $λ\in\mathbb{R}$. If the Eule…

This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature. This is a conseque…

In this note, we study the curvature flow to Nirenberg problem on $S^2$ with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed non-negative Gaussian curvature $f$ has its positive part, which possesses non-degenera…

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in …

Study on the convergence rate of prescribed scalar curvature flow.

Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with $\…

The study finds multiple conformal metrics with specific curvature properties on compact surfaces.

The paper proves a theorem for discretizing Gaussian curvature on surfaces.

Paper estimates curvature of convex hypersurfaces with prescribed curvature.

It is proved that the equality $Δ\ln|κ-λ|=6κ$, where $κ$ is the Gaussian curvature of a metric tensor g on a 2-dimensional manifold is a sufficient and necessary condition for local realizability of the metric as the Blaschke metric of some affine sphere.

In this paper, for the Lorentz manifold $M^{2}\times\mathbb{R}$, with $M^{2}$ a $2$-dimensional complete surface with nonnegative Gaussian curvature, we investigate its space-like graphs over compact strictly convex domains in $M^{2}$, which are evolving by the non-parametric mean curvature flow with prescribed contact…

Proves existence and uniqueness of Killing graphs with prescribed curvature.

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R^3, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove…

Theory proves existence of hypersurfaces with prescribed curvature.

In this work, we study graphs in $\M^n\times\Real$ that are evolving by the mean curvature flow over a bounded domain on $\M^n$, with prescribed contact angle in the boundary. We prove that solutions converge to translating surfaces in $\M^n\times\Real$. Also, for a Riemannian manifold $\M^2$ with negative Gaussian cur…

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

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

Study on prescribing Ricci curvature on compact Lie groups.

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

Proves optimal regularity for sphere minimizers in 3-sphere.

Paper solves degenerated circle pattern metric problem in spherical geometry.

Proves existence of graph on torus with prescribed curvature.

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…

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

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