arXiv research

Topological obstructions to admissibility in $σ_k$-Loewner--Nirenberg problem

Proves existence and compactness of solutions to $σ_2$-Nirenberg problem on sphere.

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…

New neural network solves Nirenberg problem for curvature on sphere.

Verified numerics prove existence of a curvature solution with known symmetries.

Study on half spheres solves Nirenberg problem with complex blow-up analysis.

The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.

This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.

The paper discusses Nirenberg's work on geometric problems and his personality.

We consider the fractional Nirenberg problem on the standard sphere $\mathbb{S}^n$ with $n\geq 4$. Using the theory of critical points at infinity, we establish an Euler-Hopf type formula and obtain some existence results for curvature satisfying assumptions of Bahri-Coron type.

Flow approach solves Ricci equation boundary problem.

Solves Loewner-Nirenberg problem on Riemannian manifolds for k ≤ n/2.

Proves existence of smooth metrics with specific curvature properties.

Two flow methods solve a problem on Riemannian manifolds, proving convergence to the Loewner-Nirenberg solution.

In this work, we develop a study involving some nonlinear partial differential equations on spheres and hemispheres, with the zero Neumann boundary condition, which are so-called Brezis-Nirenberg type problems, and we give conditions on which such equations have only constant solutions. We also extend these results for…

Study proves curvature prescription on spheres for k ≥ n/2.

The article resolves complex structures in transport twistor spaces, proving a Newlander-Nirenberg theorem.

The paper solves a Nirenberg problem on half spheres, finding multiple blow-ups.

The Nirenberg problem yields multiple conformal metrics for a given scalar curvature function.

Let (M,g) be a compact Riemannian manifold of dimension n \geq 2. In this work we prove the validity of the optimal L^p-Riemannian Gagliardo-Nirenberg inequality for 1 < p \leq 2. Our proof relies strongly on a new distance lemma which. In particular, we extend L^p-Euclidean Gagliardo-Nirenberg inequalities due to Del …

The paper proves existence of solutions to a Loewner-Nirenberg problem on Riemannian manifolds.

Solves geometric problems using fully nonlinear equations and Morse theory.

Paper solves Christoffel-Minkowski problem in hyperbolic space.

Solves nonlinear problems on metric structures through eigenvalue counting.

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with same exponent n(n>1), then it has exactly n-dimensional volume growth. As application, we obtain geometric and topological properties of Alexandrov space, Riemannian manifold …

Newlander-Nirenberg theorem extended to complex b-manifolds.

The paper solves the Nirenberg problem on high-dimensional half spheres with pinching conditions.

Solves Brezis' first open problem on ball solutions.

We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Further…

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine $L^{p}-$Sobolev inequalities. The logarithmic version of affine $L^{p}-$Sobolev inequalities is verified. Moreover, An alternative proof of the affine Mo…

The paper derives inequalities on Finsler manifolds, influenced by their curvatures.

Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations $P_σ(v)= Kv^{\frac{n+2σ}{n-2σ}}$ on the standard unit sphere $\mathbb{S}^n$ for all $σ\in (0,n/2)$, where $P_σ$ is the intertwining …

Paper proves a quantitative estimate for transforming almost complex structures into standard ones.

We study a class of compact surfaces in $\mathbb R^3$ introduced by Alexandrov and generalized by Nirenberg and prove a compactness result under suitable assumptions on induced metrics and Gauss curvatures.

We prove the concavity of $p$-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on $\mathbb{R}^n$ or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp $L^p$-Sobolev inequality and $L^p$-Gagliardo-Nirenberg inequalities on…

Maximal solution of a PDE shows boundary smoothness for certain domains.

Symmetry proven for positive solutions of a weighted p-Laplace operator inequality.

We consider the problem of finding on a given Euclidean domain $Ω$ of dimension $n \geq 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(λ(-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence a…

We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent $n \ge 3$, then it has exactly the $n$-dimensional volume growth. As an application, if an $n$-dimensional Finsler manifold of non-negative $n$-Ricci curvature satisfies th…

The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.

The paper examines the smoothness of solutions to a specific partial differential equation on smooth domains.

We consider the multi-bump solutions of the following fractional Nirenberg problem \begin{equation}\label{01} (-Δ)^s u=K(x)u^{\frac{n+2s}{n-2s}}, \;\;\;\;u>0\;\;\text{ in }\mathbb{R}^n, \end{equation} where $s\in (0,1)$ and $n>2+2s$. If $K$ is a periodic function in some $k$ variables with $1\leq k<\frac{n-2s}2$, we pr…

We prove that $n$-dimensional ($n\geqslant3$) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure $n$-space (i.e., the Euclidean metric $n$-space).

Generalizes complex manifolds to manifolds with corners and generalized corners.

The paper extends Newlander-Nirenberg theorem to domains with $C^2$ boundary.

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if…

This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $\{\begin{aligned} &Δu+ f(u)=0,\ \ \ {in}\ Ω=\{(x^\prime,x_n): x_n>\varphi (x^\prime)\},\\ &u>0,\ \ \ {in}\ Ω,\\ &u=0,\ \ \ {on}\ \partialΩ,\\ &|\nabla u|=const. {on} \partialΩ. \end{aligned}.$ We prove that up to isometry the ep…

Study extends Nirenberg-Spencer's question to families of submanifolds.