Gauss Curvature Equation Research Articles

Improved Beckner's inequality for axially symmetric functions on $\mathbb{S}^4$

We show that axially symmetric solutions on \mathbb{S}^4 to a constant Q -curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter \alpha in front of the Paneitz operator belongs to the interval [\frac{473 + \sqrt{209329}}{1800}\approx 0.517, 1) . This is in contrast to the case \alpha=1 , where there exists a family of solutions, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on \mathbb{S}^2 . As a consequence, we prove an improved Beckner's inequality on \mathbb{S}^4 for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when \alpha=1/5 by exploiting Pohozaev-type identities, and prove the existence of a non-constant axially symmetric solution for \alpha \in (1/5, 1/2) via a bifurcation method.

Remark on Gauss curvature equations on punctured disk

We give a new argument on the classification of solutions of Gauss curvature equation on ℝ2, which was first proved by W. Chen and C. Li [Duke Math. J., 1991, 63(3): 615–622]. Our argument bases on the decomposition properties of the Gauss curvature equation on the punctured disk.

Stable convergence of inner functions

Let J be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on J where F n → F if the critical structures of F n converge to the critical structure of F. We show that this occurs precisely when the critical structures of the F n are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic self-maps of the unit disk and solutions of the Gauss curvature equation. Building on the works of Korenblum and Roberts, we show that this topology also governs the behaviour of invariant subspaces of a weighted Bergman space which are generated by a single inner function.

Optimal transport and the Gauss curvature equation

In this short note, we consider the problem of prescribing the Gauss curvature and image of the Gauss map for the graph of a function over a domain in Euclidean space. The prescription of the image of the Gauss map turns this into a second boundary value problem. Our main observation is that this problem can be posed as an optimal transport problem where the target is a subset of the lower hemisphere of $\mathbb{S}^n$. As a result we obtain existence and regularity of solutions under mild assumptions on the curvature, as well as a quantitative version of a gradient blowup result due to Urbas, which turns out to fall within the optimal transport framework.

On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature

The purpose of this paper is to study the solutions of \begin{document}$ \Delta u +K(x) e^{2u} = 0 \quad{\rm in}\;\; \mathbb{R}^2 $\end{document} with \begin{document}$ K\le 0 $\end{document} . We introduce the following quantities: \begin{document}$ \alpha_p(K) = \sup\left\{\alpha \in \mathbb{R}:\, \int_{ \mathbb{R}^2} |K(x)|^p(1+|x|)^{2\alpha p+2(p-1)} dx Under the assumption \begin{document}$ ({\mathbb H}_1) $\end{document} : \begin{document}$ \alpha_p(K)> -\infty $\end{document} for some \begin{document}$ p>1 $\end{document} and \begin{document}$ \alpha_1(K) > 0 $\end{document} , we show that for any \begin{document}$ 0 , there is a unique solution \begin{document}$ u_\alpha $\end{document} with \begin{document}$ u_\alpha(x) = \alpha \ln |x|+ c_\alpha+o\big(|x|^{-\frac{2\beta}{1+2\beta}} \big) $\end{document} at infinity and \begin{document}$ \beta\in (0, \, \alpha_1(K)-\alpha) $\end{document} . Furthermore, we show an example \begin{document}$ K_0 \leq 0 $\end{document} such that \begin{document}$ \alpha_p(K_0) = -\infty $\end{document} for any \begin{document}$ p>1 $\end{document} and \begin{document}$ \alpha_1(K_0) > 0 $\end{document} , for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution \begin{document}$ u_{\alpha_*} $\end{document} such that \begin{document}$ u_{\alpha_*} -\alpha_*\ln|x| = O(1) $\end{document} at infinity for some \begin{document}$ \alpha_* > 0 $\end{document} , which does not converge to a constant at infinity. This example exhibits a new phenomenon of solution with logarithmic growth, finite total curvature, and non-uniform asymptotic behavior at infinity.

Strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms

This article is devoted to C2 a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature equation under the assumption of a strictly locally convex subsolution.

Nondegeneracy of the Gauss curvature equation with negative conic singularity

We study the Gauss curvature equation with negative singularities. For a local mean field type equation with only one negative index we prove a uniqueness property. For a global equation with one or two negative indexes we prove the non-degeneracy of the linearized equations.

Vanishing Estimates for Fully Bubbling Solutions of SU (n + 1) Toda Systems at a Singular Source

AbstractFor Gauss curvature equation (or more general Toda systems) defined on 2D spaces, the vanishing rate of certain curvature functions on blowup points is a key estimate for numerous applications. However, if these equations have singular sources, very few vanishing estimates can be found. In this article we consider a Toda system with singular sources defined on a Riemann surface and we prove a very surprising vanishing estimates and a reflection phenomenon for certain functions involving the Gauss curvature.

Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature

We consider the numerical approximation of surfaces of prescribed Gaussian curvature via the solution of a fully nonlinear partial differential equation of Monge-Ampere type. These surfaces need not be continuous up to the boundary of the domain and the Dirichlet boundary condition must be interpreted in a weak sense. As a consequence, sub-solutions do not always lie below super-solutions, standard comparison principles fail, and existing convergence theorems break down. By relying on a geometric interpretation of weak solutions, we prove a relaxed comparison principle that applies only in the interior of the domain. We provide a general framework for proving existence and stability results for consistent, monotone finite difference approximations and modify the Barles-Souganidis convergence framework to show convergence in the interior of the domain. We describe a convergent scheme for the prescribed Gaussian curvature equation and present several challenging examples to validate these results.

Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation on the torus

In the spirit of the previous paper (Borer et al., Commun Math Helv, 2015), where we dealt with the case of a closed Riemann surface $$(M,g_0)$$ of genus greater than one, here we study the behaviour of the conformal metrics $$g_\lambda $$ of prescribed Gauss curvature $$K_{g_\lambda } = f_0 + \lambda $$ on the torus, when the parameter $$\lambda $$ tends to one of the boundary points of the interval of existence of $$g_\lambda $$ , and we characterize their “bubbling behavior” as in Borer et al. (Commun Math Helv, 2015).

Global singularity theory for the Gauss curvature equation

Lecture notes for a minicourse to given in the XVII Brazilian School of Geometry, UFAM (Amazonas), Brazil, July 2012.

Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature

In this article we investigate a general class of Monge-Ampere equations in the plane, including the constant Gauss curvature equation. Our first aim is to prove some maximum and minimum principles for suitable $P$-functions, in the sense of L.E. Payne. Then, these new principles are employed to solve a general class of overdetermined Monge-Ampere problems and to investigate two boundary value problems for the constant Gauss curvature equation. More precisely, when the constant Gauss curvature equation is subject to the homogeneous Dirichlet boundary condition, we prove several isoperimetric inequalities, while when it is subject to the contact angle boundary condition, some necessary conditions of solvability, involving the curvature of the boundary of the underlying domain and the given contact angle, are derived.

A Liouville theorem for conformal Gaussian curvature type equations in $${{\mathbb{R}}^2}$$

In this paper, we obtain a Liouville type theorem for a class of elliptic equations including the conformal Gaussian curvature equation $$-\Delta u=K(x)e^{2u}\quad {\rm in}\,\, {\mathbb{R}}^2,$$ where K(x) is a Holder continuous function in \({{\mathbb{R}}^2}\) that does not have a fixed sign near infinity. The main tool in our approach is an asymptotic formula for the solution at infinity and the method of moving planes. We also show how our Liouville theorem can be used to obtain a priori bound for solutions of the prescribing Gaussian curvature equation in S2, namely $$\Delta\, u+K(x)e^{2u}=1\, {\rm in}\, S^2,$$ where K(x) is Holder continuous and nonnegative in S2 but vanishes on a set with nonempty interior, a case left open in previous research.

The obstacle problem for Monge-Ampère type equations in non-convex domains

In this paper, we consider the obstacle problem for Monge-Ampere type equations which include prescribed Gauss curvature equation as a special case. We establish $C^{1,1}$ regularity of the greatest viscosity solution in non-convex domains.

The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

AbstractA classical result of Nitsche [22] about the behaviour of the solutions to the Liouville equation Δu= 4e2unear isolated singularities is generalized to solutions of the Gaussian curvature equation Δu= −κ(z)e2uwhere κ is a negative Hölder continuous function. As an application a higher–order version of the Yau–Ahlfors–Schwarz lemma for complete conformal Riemannian metrics is obtained.

On the isolated singularities of the solutions of the Gaussian curvature equation for nonnegative curvature

The precise asymptotic behaviour of the solutions to the two-dimensional curvature equation Δ u = k ( z ) e 2 u with e 2 u ∈ L 1 for bounded nonnegative curvature functions − k ( z ) near isolated singularities is obtained.

A priori estimate for the Nirenberg problem

We establish a priori estimate for solutions to the prescribing Gaussian curvature equation <p align="center"> $ - \Delta u + 1 = K(x) e^{2u}, x \in S^2,$ (1) <p align="left" class="times"> for functions $K(x)$ which are allowed to change signs. In [16], Chang, Gursky and Yang obtained a priori estimate for the solution of (1) under the condition that the function K(x) be positive and bounded away from 0. This technical assumption was used to guarantee a uniform bound on the energy of the solutions. The main objective of our paper is to remove this well-known assumption. Using the method of moving planes in a local way, we are able to control the growth of the solutions in the region where K is negative and in the region where K is small and thus obtain a priori estimate on the solutions of (1) for general functions K with changing signs.

Generalized solutions of curvature equations

We introduce a notion of “generalized solutions” for the so-called curvature equations. It is an extension of the definition of generalized solutions of the Gauss curvature equation and gives a new framework for the study of more general curvature equations. We prove that if the curvature equation has a convex solution, then the inhomogeneous term must be a Borel measure. We also discuss basic properties of generalized solutions. Among other things, we show that our class of generalized solutions is wider than that of weak solutions.

Regularity of Degenerate Monge–Ampère and Prescribed Gaussian Curvature Equations in Two Dimensions

We use a priori inequalities for quasilinear equations to obtain a regularity theorem for the Dirichlet problem for the Monge–Ampere equation, $$u_{xx}u_{yy}-(u_{xy})^{2}=k(x,y),$$ and the prescribed Gaussian curvature equation, $$u_{xx}u_{yy}-(u_{xy})^{2}=k(x,y)(1+u_{x}^{2}+u_{y}^{2})^{2},$$ where k(x,y) is close to a function of one variable alone when k is small, but permitted to vanish to infinite order.

Uniqueness and non-existence theorems for conformally invariant equations

By using the equivalent integral form for the Q -curvature equation, we generalize the well-known non-existence results on two-dimensional Gaussian curvature equation to all dimensional Q -curvature equation. Somehow, we introduce a new approach to Q -curvature equation which is higher order and even pseudo-differential equation. As a by-product, we do classify the solutions for Q = 1 solutions on S n as well as on R n with necessary growth rate assumption.