arXiv research

Sharp bound on scalar curvature integral in 3-manifolds.

Global regularity proved for 4D Ricci flow with scalar curvature integral bound.

Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.

Study on scalar curvature deformations in pseudohermitian manifolds.

Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.

New bounds on scalar curvature for metric sequences.

Calabi flow extended with bounded curvature integrals.

The paper explores geometry and positive scalar curvature on non-compact manifolds.

New method solves complex curvature equations.

The paper improves curvature estimates for Ricci flow solutions with bounded scalar curvature.

Upper diameter bound for manifolds with positive scalar curvature.

Proves Ricci flow extensibility with integral norms.

Study on gradient h-almost Yamabe solitons with scalar curvature estimation.

Lower bounds on curvature integral for manifolds with curvature constraints.

In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with no…

Using the monotonicity formulas of Colding and Minicozzi, we prove that on any complete, non-parabolic Riemannian manifold $(M^3, g)$ with non-negative Ricci curvature, the asymptotic weighted scaling invariant integral of scalar curvature has an explicit bound in form of asymptotic volume ratio.

Study torsion obstructions to positive scalar curvature on manifolds.

Deform quantization recovers scalar curvature in complex structures.

We compute renormalized curvature integrals on Poincaré-Einstein manifolds.

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

In this paper we prove that, under an explicit integral pinching assumption between the $L^2$-norm of the Ricci curvature and the $L^2$-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits an Einstein metric with positive curvature. In particular this implies that the manifold is diff…

The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.

We prove a regularity result for unit volume conformal metrics with integral scalar curvature bounds for $p>n/2$ and first eigenvalue of $Δ$ bounded from below by a constant $B > Λ_1(S^n,[g_{st.}]).$

We present in this note a lower bound for the Calabi functional in a given Kähler class. This yields an integral inequality for constant scalar curvature metrics, which can be viewed as a refined version of Yau's Chern number inequality.

We prove a priori interior curvature estimates for hypersurfaces of prescribing scalar curvature equations in dimension three. The method is motivated by the integral method of Warren and Yuan. The new observation here is that the "Lagrangian" submanifold constructed similarly as Harvey and Lawson has bounded mean curv…

The paper studies $m$-quasi Einstein manifolds with convex potential and finds constant scalar curvature.

Study short-time existence of Ricci-DeTurck flow from rough metrics with Morrey-type integrability.

The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Ch…

Derives inequalities for eigenvalues and renormalized volume of Poincaré-Einstein manifolds.

We apply conformal flows of metrics restricted to the orthogonal distribution $D$ of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of $D$, and for the case …

We study the existence of a metric with zero scalar curvature maximizing the isoperimetric ratio among all zero scalar curvature metrics in a fixed conformal class of metrics on a compact manifold with boundary. The question may be reduced to an extremal problem for the harmonic extension of functions and the related n…

Using spin$^c$ structure we prove that Kähler-Einstein metrics with nonpositive scalar curvature are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Moreover if all infinitesimal complex deformation of the complex structure are integrable, th…

Study on biharmonic hypersurfaces in spheres and space forms, proving rigidity under scalar curvature condition.

Geometrically represents path integral reduction Jacobian for interacting systems.

Let $M^n(n\geq3)$ be an $n$-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $M^n$ satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds with harmonic curvature and positive scalar curvature. In particular, Theorem 1.4,…

Study on ALH manifolds with boundary, showing surjectivity of scalar curvature map and mass rigidity.

Based on Donaldson's method, we prove that, for an integral Kahler class, when there is a Kahler metric of constant scalar curvature, then it minimizes the K-energy. We do not assume that the automorphism group is discrete.

We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of definin…

The study characterizes hypersurfaces in spheres with constant scalar curvature.

On a manifold $(\mathbb{R}^n, e^{2u} |dx|^2)$, we say $u$ is normal if the $Q$-curvature equation that $u$ satisfies $(-Δ)^{\frac{n}{2}} u = Q_g e^{nu}$ can be written as the integral form $u(x)=\frac{1}{c_n}\int_{\mathbb R^n}\log\frac{|y|}{|x-y|}Q_g(y)e^{nu(y)}dy+C$. In this paper, we show that the integrability assum…

In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bo…

The paper studies Ricci flows with bounded scalar curvature and proves convergence to orbifolds.

Let (M,J) be a compact complex 2-manifold which which admits a Kaehler metric for which the integral of the scalar curvature is non-negative. Also suppose that M does not admit a Ricci-flat Kähler metric. Then if M is blown up at sufficiently many points, the resulting complex surface admits Kaehler metrics with scalar…

The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.

We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the $L^2$-norm of their scalar curvature and…

New rigidity theorems for spin fill-ins with non-negative scalar curvature.

We derive lower bounds on the scalar curvature of complete non-compact gradient Yamabe solitons under some integral curvature conditions. Based on this, we prove that the corresponding potential functions have at most quadratic growth in distance. We also obtain a finite topological type property on complete shrinking …

In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the same integral Bochner technique to extend the results in dimension three. Then, by using the classification of closed thre…