arXiv research

The study preserves upper bounds of total scalar curvature in conformal classes.

Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.

The paper examines compactness of scalar curvature sequences on conformal manifolds.

In the class of metrics of a generic conformal structure there exists a distinguishing metric. This was noticed by Albert Einstein in a lesser-known paper of 1921 (Berl. Ber., 1921, pp. 261-264). We explore this finding from a geometrical point of view. Then, we obtain a family of scalar conformal invariants of weight …

We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We als…

The paper tackles scalar curvature on manifolds conformal to tori, proving stability under certain conditions.

Develops methods for computing conformal invariants of submanifolds.

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

Study negative scalar curvature metrics with positive boundary mean curvature.

New examples show positive scalar curvature metrics on manifolds with boundary that cannot be extended.

In this article, we found a connection between Brown-York mass and the first Dirichlet Eigenvalue of a Schrödingier type operator. In particular, we proved a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigated compactly conformal defo…

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

We study here compact manifolds with positive scalar curvature metrics. We use the relative Yamabe invariant from math.DG/0008138 to define the conformal cobordism relation on the category of such manifolds. We prove that corresponding conformal cobordism groups $\Pos_n^{\conf}(γ)$ are isomorphic to the cobordism group…

The paper examines Randers metrics with isotropic scalar curvature properties.

Conditions for scalar curvature on compact manifolds under conformal deformation.

The study proves a new upper bound for isoperimetric ratio in scalar-flat conformal classes.

The paper solves a problem related to curvature in complex geometry.

Extends moment map concept to locally conformally Kähler manifolds.

Conformal invariants of manifolds of non-positive scalar curvature are studied in association with growth in volume and fundamental group.

Conformally variational Riemannian invariants (CVIs), such as the scalar curvature, are homogeneous scalar invariants which arise as the gradient of a Riemannian functional. We establish a wide range of stability and rigidity results involving CVIs, generalizing many such results for the scalar curvature.

Study shows incompatibility of certain scalar curvatures on manifolds.

Proves properties of 4-manifolds with scalar curvature constraints.

The article contains a brief description on the study of conformal scalar curvature equations, and discusses selected topics and questions concerning the equations in open spaces.

Study curvatures on compact pseudo-Hermitian manifolds using special methods.

Optimal controls for conformal Laplacian obstacle problems on spheres and manifolds.

The paper explores constant holomorphic d-scalar curvature on specific manifolds.

Unique conformal metrics found on certain manifolds.

Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.

Constructs metrics with constant scalar curvature and unbounded volumes on spheres.

We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. We also give a new conformally invariant …

The paper studies conformally flat cubic metrics with isotropic curvature, finding they must be Minkowski.

We compute renormalized curvature integrals on Poincaré-Einstein manifolds.

We discuss conformal deformation and warped products on some open manifolds. We discuss how these can be applied to construct Riemannian metrics with specific scalar curvature functions.

Researchers solve the negative Yamabe case for scalar curvature prescription.

We consider the conformal class of the Riemannian product $g_0 + g$, where $g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect…

We survey some results on scalar curvature and properties of solutions to the Einstein constraint equations. Topics include an extended discussion of asymptotically flat solutions to the constraint equations, including recent results on the geometry of the center of mass of such solutions. We also review methods to con…

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n\geq 3$. We prove the existence of such conformal metrics in the cases of $n=6,7$ or the manifold is spin and some other remai…

In this paper, we study conformally flat hypersurfaces of dimension $n(\geq 4)$ in $\mathbb{S}^{n+1}$ using the framework of Möbius geometry. First, we classify and explicitly express the conformally flat hypersurfaces of dimension $n(\geq 4)$ with constant Möbius scalar curvature under the Möbius transformation group …

Study non-positive scalar curvature on hyperbolic manifolds, finding necessary and sufficient conditions.

New operators and curvatures derived from embedded manifolds.

Researchers find unique metrics solving complex PDEs for constant scalar curvature.

Compact metrics found with specific curvature properties on 3D surfaces.

Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on $M$, with an appropriate control on the Ricci curvature makes $M$ to be isometric to a hemisphere of $\mathbb{S}^{n}$. We also prove that if an Ein…

Study properties of hypersurfaces in spacetimes with conformal transformations.

We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yam…

Simplified Obata-Vétois argument for Einstein manifolds with nonnegative scalar curvature.

Let $(M^{n},g_{0})$ be a $n=3,4,5$ dimensional, closed Riemannian manifold of positive Yamabe invariant. For a smooth function $K>0$ on $M$ we consider a scalar curvature flow, that tends to prescribe $K$ as the scalar curvature of a metric $g$ conformal to $g_{0}$. We show global existence and in case $M$ is not confo…

Let (M,g) be a compact Riemannian manifold with dimension n > 2. The Yamabe problem is to find a metric with constant scalar curvature in the conformal class of g, by minimizing the total scalar curvature. The proof was completed in 1984. Suppose (M',g') and (M'',g'') are compact Riemannian n-manifolds with constant sc…