arXiv research

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

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

Proves existence of smooth metrics with specific curvature properties.

Flow approach solves Ricci equation boundary problem.

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

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

Solves geometric problems using fully nonlinear equations and Morse theory.

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

Solves nonlinear problems on metric structures through eigenvalue counting.

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…

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

The paper classifies solutions to a specific equation and finds counterexamples to boundary estimates.

This paper carries out a renormalization of the volume of the Loewner-Nirenberg singular Yamabe metric in a given conformal class on a compact manifold-with-boundary. This generalizes the usual volume renormalization for Poincare-Einstein metrics. The coefficient of the log term in the volume expansion defines a confor…

We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first resu…

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…

Study on metrics on manifolds with specific curvature properties.

Study properties of solutions with singularities in the negative cone.

We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic of a four dimensional manifold-with-boundary in terms of the geometry of the Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the…

The negative case of the Singular Yamabe Problem concerns the existence and behavior of complete metrics with constant negative scalar curvature on the complement of a closed set in a compact Riemannian manifold which are conformally equivalent to a smooth metric on this compact manifold. When the closed set is a smoot…

The paper proves Ricci flow convergence on compact manifolds with boundary.

The paper extends a Ricci flow result for compact manifolds with boundary.