arXiv research

Study solves Yamabe problems on metric measure spaces with or without boundary.

Solve supercritical Yamabe problem on manifolds with non-umbilic boundary.

The paper examines stability of Yamabe boundary problem under perturbations.

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type m…

The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.

New scalars measure failure of CC metrics to solve singular Yamabe problem.

Study asymptotic behaviors of solutions near singular boundaries for the Yamabe problem.

Study on compactness and blow-up of solutions for Yamabe problems on manifolds with non-umbilic boundaries.

Solutions grow for a special type of math problem on curved spaces.

In this paper, we solve the remaining cases of the boundary Yamabe problem introduced by Escobar in 1992. Indeed, using the bubbles of Brendle-Chen, which are an adaptation to manifolds with boundary of the original ones introduced by Brendle for the study of the Yamabe flow on closed Riemannian manifolds of dimension …

New local method solves Yamabe problems on compact and non-compact manifolds.

In this paper, we study multiplicity of solutions of the Yamabe problem on product manifolds with minimal boundary via bifurcation theory.

Solves boundary Yamabe problem with minimal boundary scenario.

This paper concerns a fully nonlinear version of the Yamabe problem on manifolds with boundary. We establish some existence results and estimates of solutions.

We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with umbilic boundary, provided the Weyl tensor is nonzero everywhere on the boundary and the dimension of the manifold is n>10.

We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n>6 and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.

Study finds least-energy nodal solutions to the Yamabe problem on manifolds with boundary.

In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.

The fractional Yamabe problem, proposed by González-Qing (2013, Anal. PDE) is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal co…

The paper proves stability for scalar-flat metrics on manifolds with boundary.

The paper finds sign-changing solutions for a specific type of elliptic equation.

Paper studies flow on hyperbolic surfaces to match boundary lengths.

A new flow method solves the weighted Yamabe problem with boundary.

Paper proves uniqueness of Type II Yamabe metrics on manifolds.

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold endowed with the Webster metric hence formulate a version of the CR Yamabe problem for CR manifolds-with-boundary. This is shown to be a nonlinear subelliptic problem of variational origin.

Paper examines stability of minimizing metrics on manifolds with boundary.

Solves a complex mathematical problem on curved surfaces.

We define a relative Yamabe invariant of a smooth manifold with given conformal class on its boundary. In the case of empty boundary the invariant coincides with the classic Yamabe invariant. We develop approximation technique which leads to gluing theorems of two manifolds along their boundaries for the relative Yamab…

The paper proves solutions for Yamabe equations on manifolds with boundary.

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing $σ_k$-curvature in the interior and constant $H_k$-curvature on the boundary. When restricting to the closure of the positive $k$-cone, this is…

Study finds infinitely many non-radial solutions for negative scalar curvature in higher dimensions.

Study of scattering on singular Yamabe spaces using asymptotically hyperbolic manifolds.

Solves Neumann problem on CR manifold boundary.

Study on solutions to spinorial Yamabe equation on manifolds with boundary.

Study on metrics on manifolds with specific curvature properties.

This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The combinatorial Yamabe flow is a gradient flow of a concave function. The long ti…

The Willmore energy, alias bending energy or rigid string action, and its variation-the Willmore invariant-are important surface conformal invariants with applications ranging from cell membranes to the entanglement entropy in quantum gravity. In work of Andersson, Chrusciel, and Friedrich, the same invariant arises as…

Let M,g a compact Riemannian n-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this …

We consider a nonlinear version of the Yamabe problem on locally conformally flat compact manifolds with boundary. The main technique we used is to derive boundary $C^2$ estimates directly from boundary $C^0$ estimates. In particular, the result is a generalization of the work by Escobar.

Generalized stability theorem for compact manifolds with boundary.

The paper extends a connected-sum inequality to calculate λ-Yamabe invariants of certain manifolds.

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -Δ_{g}u+au=0 & \text{ on }M \\ \partial_νu+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(…

The paper solves scalar curvature problems under conformal deformation for Riemannian manifolds.

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric $g$, there is a conformally equivalent asymptotically flat scal…

We study the Yamabe flow on compact Riemannian manifolds of dimensions greater than two with minimal boundary. Convergence to a metric with constant scalar curvature and minimal boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin.

Study of conformally compact metrics and Lovelock tensors in even dimensions.

Let (M,g) be a compact Riemannian manifold with boundary. This paper addresses the Yamabe-type problem of finding a conformal scalar-flat metric on M, which has the boundary as a constant mean curvature hypersurface. When the boundary is umbilic, we prove an existence theorem that finishes some remaining cases of this …

Paper finds conditions for non-Einstein relative Yamabe metrics.