arXiv research

In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.

The paper shows integrability of scalar curvature implies normal metric on conformally flat manifolds.

New inequalities for austere submanifolds established.

We study a pointwise inequality for submanifolds in real space forms involving the scalar curvature, the normal scalar curvature and the mean curvature. We translate it into an algebraic problem, allowing us to prove a slightly weaker version of it. We also prove the conjecture for certain types of submanifolds of $\ma…

In this paper, we proved the Normal Scalar Curvature Conjecture and the Bottcher-Wenzel Conjecture. We also established some new pinching theorems for minimal submanifolds in spheres.

Totally geodesic submanifolds in spheres have restricted curvature properties.

This paper determines bounds on normal scalar curvature of isoparametric hypersurface focal submanifolds.

The paper examines flatness conditions on normal metric contact pairs and proves properties of Einstein manifolds.

We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kahler surfaces, projective manifolds o…

We investigate the scalar curvature behavior along the normalized conical Kähler-Ricci flow $ω_t$, which is the conic version of the normalized Kähler-Ricci flow, with finite maximal existence time $T<\infty$. We prove that the scalar curvature of $ω_t$ is bounded from above by $C/(T-t)^2$ under the existence of a con…

This paper studies normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically -1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature -1. A relative estimate of Green's function is proved as a tool.

The paper studies a modified scalar curvature flow and proves convergence to a sphere.

Paper proves estimates for metrics with conic singularities.

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

Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension

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…

The paper proves inequalities for scalar curvature on various manifolds.

The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.

In this paper, we give a proof of the DDVV conjecture which is a pointwise inequality involving the scalar curvature, the normal scalar curvature and the mean curvature on a submanifold of a real space form. Furthermore we solved the problem of its equality case.

Paper proves properties of minimal hypersurfaces in specific solitons.

Let D a divisor with simple normal crossings in a Kahler manifold X. The purpose of this short note is to show that the existence of a Poincare type metric with constant scalar curvature in on the complement of D implies for any component of the divisor that the scalar curvature of Poincare type metric outside of D is …

The paper extends Yamabe flow results to non-compact manifolds with bounded geometry.

Study shows scalar curvature of a specific type of manifold converges to -m outside singular points.

Researchers prove spaces of positive scalar curvature metrics are contractible with symmetry.

Study existence of conformal metrics with specific curvature properties on compact manifolds.

Study curvatures of submanifolds in space forms with topological obstructions.

Proves a conjecture about manifolds and scalar curvature.

The paper proves curvature inequalities for submanifolds in space forms.

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

In this paper we prove two sharp inequalities involving the normalized scalar curvature and the generalized normalized $δ$-Casorati curvatures for slant submanifolds in quaternionic space forms. We also characterize those submanifolds for which the equality cases hold. These results are a generalization of some recent …

The paper finds optimal inequalities for hypersurface curvatures in complex Grassmannians.

Sharp inequality for complex projective space's systoles under scalar curvature bounds.

The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we d…

Study self-expanding solutions of mean curvature flow in various dimensions.

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0>σ_{n}K_{\max}$, where $σ_n\in (\frac{1}{4},1)$ is an explicit positive constan…

Researchers solve curvature prescription for foliated Riemann-Cartan manifolds.

Paper shows triviality criterion for certain Ricci solitons.

In this note we give a survey on the DDVV conjecture which is also called the "normal scalar curvature conjecture".

New stable metric found on a complex space.

In this paper, we use the normalized Ricci-DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature $R\geq -n\left(n-1\right)$ and also the rigidity result when certain …

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

Consider a compact Kähler manifold X with a simple normal crossing divisor D, and define Poincaré type metrics on X\D as Kähler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincaré type Kähler metric on X\D implies the existence of a con…

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

A new flow method reduces Lorentz contraction to a simple algebraic decay.

Given a submanifold $S \subset \mathbb R^n$ of codimension at least three, we construct an asymptotically Euclidean Riemannian metric on $\mathbb R^n$ with nonnegative scalar curvature for which the outermost apparent horizon is diffeomorphic to the unit normal bundle of $S$.

The study pinches rigidity theorems for minimal submanifolds in spheres.

In this paper we characterize logarithmic surfaces which admit Kähler-Einstein metrics with negative scalar curvature and small edge singularities along a normal crossing divisor.

Given a hypersurface $M$ of null scalar curvature in the unit sphere $\mathbb{S}^n$, $n\ge 4$, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in $\Rr^{n+1}$ as a normal graph over a truncated cone generated by $M$. Furthermore, this graph is 1-stable if t…