arXiv research

The paper examines complete Yamabe solitons with finite total scalar curvature.

The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.

In this article we show that any finite cover of the moduli space of closed Riemann surfaces of $g$ genus with $g\geq 2$ does not admit any complete finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we also show that the total mass of the scalar curvature of any almost Hermitian metric, which i…

Revisits conformal metrics with finite Q-curvature, providing necessary and sufficient conditions.

Characterizes metrics with finite total Q-curvature and introduces new volume entropy.

We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface $M$ in $\mathbb{R}^{4}$ with zero scalar curvature $S_2$, nonzero Gauss-Kronecker…

In this note, we give a geometric characterization of the compact and totally umbilical hypersurfaces that carry a non trivial locally static Killing Initial Data (KID). More precisely, such compact hypersurfaces have constant mean curvature and are isometric to one of the following manifolds: (i) Sn the standard spher…

Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.

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

Totally geodesic hypersurfaces in a sphere have small total curvature.

We study the supremum of the total mean curvature on the boundary of compact, mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed boundary metric. We establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained. When the boundary consis…

Totally geodesic minimal hypersurfaces in $\mathbb H^5$ with specific curvature properties.

The study preserves lower bounds of total scalar curvature under specific metric convergence.

Paper proves conjecture about critical metrics with divergence-free Bach tensor.

Study on nonspin manifolds with spin boundary, showing nonconnectedness and nontrivial fundamental group.

The paper proves gap properties for critical metrics under specific conditions.

We solve the modified Kazdan-Warner problem of finding metrics with prescribed scalar curvature and unit total volume.

We show that a complete $m$-dimensional immersed submanifold $M$ of $\mathbb{R}^{n}$ with $a(M)<1$ is properly immersed and have finite topology, where $a(M)\in [0,\infty]$ is an scaling invariant number that gives the rate that the norm of the second fundamental form decays to zero at infinity. The class of submanifol…

The paper characterizes D'Atri spaces using total scalar curvature of hemispheres.

Totally geodesic submanifolds in spheres have restricted curvature properties.

Constructs metrics for surfaces to show uniform positive scalar curvature.

We consider an asymptotically flat Riemannian spin manifold of positive scalar curvature. An inequality is derived which bounds the Riemann tensor in terms of the total mass and quantifies in which sense curvature must become small when the total mass tends to zero.

Study bounds total mean curvature of fill-ins with scalar curvature constraints.

Directly applies Kazdan--Warner results to prescribe scalar curvature on bundles.

The paper proves uniformization for specific curvature types on manifolds.

On a compact n-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1984 by Besse, but has yet to be proved. In this paper, we prove th…

Proves properties of 4-manifolds with scalar curvature constraints.

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a Kähler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the n…

Study on totally real submanifolds in $(LCS)_n$-manifolds.

The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in $1980$'s that every CPE metric must be Einstein. We prove that a $4$-dimensional CPE…

Constructs infinitely many examples of large manifolds with circle bundles of positive scalar curvature.

It is a basic tenet in complex geometry that {\it negative} curvature corresponds, in a suitable sense, to the absence of rational curves on, say, a complex projective manifold, while {\it positive} curvature corresponds to the abundance of rational curves. In this spirit, we prove in this note that a projective manifo…

We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space …

Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.

On a compact $n$-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature o…

New Kazdan-Warner problem for equivariant metrics on manifolds.

The authors showed in a preceding paper that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed on…

It has recently been conjectured that the eigenvalues $λ$ of the Dirac operator on a closed Riemannian spin manifold $M$ of dimension $n\ge 3$ can be estimated from below by the total scalar curvature: $λ^2 \ge \frac{n}{4(n-1)} \cdot \frac{\int_M S}{vol(M)}.$ We show by example that such an estimate is impossible.

The study finds the minimum average area ratio on hyperbolic manifolds and its relation to scalar curvature.

Ancient flows converge fast with finite curvature and convexity.

In this paper we will perturb the scalar curvature of compact Kahler manifolds by incorporating it with higher Chern forms, and then show that the perturbed scalar curvature has many common properties with the unperturbed scalar curvature. In particular the perturbed scalar curvature becomes a moment map, with respect …

Study minimal surfaces with finite curvature in a specific manifold.

Study of prescribing scalar curvature and mean curvature on compact manifolds with boundary.

The study examines scalar curvature in direct product Riemannian manifolds and their multiconformal classes.

In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the following celebrated result: the Ricci curvature uniquely determines the Levi-Civita…

Geometrically represents the Jacobian for a mechanical system with symmetry.

Estimates Bartnik mass for metrics with nonnegative Gauss curvature.

We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the si…