arXiv research

Study on a specific obstruction in four-dimensional geometry.

We develop the calculus for hypersurface variations based on variation of the hypersurface defining function. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the obstruction to smoothly solving the singular Yamabe problem for conformally compac…

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

Study on scalar curvature in wedge spaces with existence and obstruction results.

We compute the Yamabe invariants for a new infinite class of closed $4$-dimensional manifolds by using a "twisted" version of the Seiberg-Witten equations, the $\mathrm{Pin}^-(2)$-monopole equations. The same technique also provides a new obstruction to the existence of Einstein metrics or long-time solutions of the no…

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

Researchers define residue families and use them to solve singular Yamabe problems.

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…

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…

Solves geometric problems using fully nonlinear equations and Morse theory.

Solves nonlinear problems on metric structures through eigenvalue counting.

On a smooth closed oriented 4-manifold $M$ with a smooth action by a finite group $G$, we show that a $G$-monopole class gives the $L^2$-estimate of the Ricci curvature of a $G$-invariant Riemannian metric, and derive a topological obstruction to the existence of a $G$-invariant nonsingular solution to the normalized R…

Renormalized volume invariant for knots in 3-sphere computed.

The aim of this paper is to study new classes of Riemannian manifolds endowed with a smooth potential function, including in a general framework classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and, above all, gradient Ricci solitons. For the most rigid cases we give a complete cl…

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 develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. Fo…

Study CR Yamabe constant, flow, and soliton on CR manifolds.

Study on convergence rate of weighted Yamabe flow.

The paper classifies a type of solitons in Euclidean spaces.

Study properties of hyperbolic Yamabe solitons in submanifolds.

The Yamabe flow converges to a specific function on compactified manifolds.

Study on $η$-Ricci-Yamabe solitons on Riemannian submersions.

Paper examines properties of specific solutions to Yamabe flow.

The study classifies h-almost Ricci-Yamabe solitons in various paracontact manifolds.

In this paper we initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds whose soliton fields are the tangential components of their position vector fields. Several fundamental results of such solitons were proved. In particular, we classify such Yamabe and quasi-Yamabe solitons on Euclidean hy…

Paper proves uniqueness of Type II Yamabe metrics on manifolds.

Paper finds conditions for non-Einstein relative Yamabe metrics.

Study CR Yamabe constant and CR structures on manifolds.

A closed CR 3-manifold is said to have $C_{0}$-positive pseudohermitian curvature if $(W+C_{0}Tor)(X,X)>0$ for any $0\neq X\in T_{1,0}(M)$. We discover an obstruction for a closed CR 3-manifold to possess $C_{0}$-positive pseudohermitian curvature. We classify closed three-dimensional CR Yamabe solitons according to $C…

The paper classifies 3D complete gradient Yamabe solitons.

Paper shows k-Yamabe solitons have constant curvature under certain conditions.

Study convergence of Yamabe flow on singular spaces with positive constant.

The study compares and finds Yamabe constants on warped products.

The paper studies Yamabe metrics and stability in Riemannian manifolds.

Quantitative stability for nearly minimizing Yamabe metrics.

The study characterizes quasi Yamabe solitons with potential vector fields.

Characterizes $*$-$k$-Ricci-Yamabe solitons on Kenmotsu manifolds.

We consider the equivariant Yamabe problem, i.e. the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their …

The $(α,β)$-Ricci-Yamabe flow exists on closed manifolds.

Study on Yamabe solitons with applications and structure elucidation.

In this paper, we completely classify almost Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field. Some results of almost Yamabe solitons with a concurrent vector field and almost Yamabe solitons on submanifolds in Riemannian manifolds equipped with a concurrent vector field are al…

Proves product metrics are Yamabe metrics under small flat torus conditions.

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

We provide the classification of locally conformally flat gradient Yamabe solitons with positive sectional curvature. We first show that locally conformally flat gradient Yamabe solitons with positive sectional curvature have to be rotationally symmetric and then give the classification and asymptotic behavior of all r…

Study shows long-term flow on special manifolds with positive Yamabe constant.

Paper examines stability of minimizing metrics on manifolds with boundary.

We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…

This paper classifies solitons under specific tensor conditions.