Sharp bound on scalar curvature integral in 3-manifolds.
problem Bounding the integral of scalar curvature on 3-manifolds.
method Geodesic ball analysis with nonnegative Ricci curvature.
result Integral of scalar curvature is bounded by 8πR for large radii. Lower bounds on curvature integral for manifolds with curvature constraints.
problem Bounding curvature integrals under curvature constraints.
method Proving a lower bound for the curvature integral using dimension, upper curvature bounds, and injectivity radius.
result Uniformly bounded below integral of scalar curvature.
Study compares isoperimetric profiles on manifolds with integral Ricci curvature bounds.
problem Comparing isoperimetric profiles on manifolds with integral Ricci curvature bounds.
method Extending previous work, the study uses integral bounds on Ricci curvature to prove comparison results for isoperimetric profile functions.
result Comparison results for the Isoperimetric profile function in manifolds with integral bounds on Ricci curvature.
Global regularity proved for 4D Ricci flow with scalar curvature integral bound.
problem Global regularity of 4D Ricci flow with integral scalar curvature bound.
method Extended Ge-Jiang's result to include integral bound on scalar curvature.
result Global ε-regularity for 4D Ricci flow with integral scalar curvature bound. Sharp spectral gap estimates on manifolds with integral curvature bounds.
problem Proving spectral gap estimates on manifolds with integral curvature bounds.
method Generalizing previous results to include integral curvature bounds.
result Confirms a conjecture about spectral gap estimates on manifolds with integral curvature bounds.
The paper improves curvature estimates for Ricci flow solutions with bounded scalar curvature.
problem Proving curvature estimates for Ricci flow solutions with bounded scalar curvature.
method Localised weighted curvature integral estimates for solutions to Ricci flow.
result Integral curvature estimates imply a uniform bound on the spatial L2 norm of the Riemannian curvature tensor. Sharp lower bound found for integral varifolds' mean curvature.
problem Finding a sharp lower bound for the mean curvature integral of integral varifolds.
method Developed a new approach using integral varifolds and mean curvature.
result A sharp lower bound on the mean curvature integral with critical power for integral varifolds.
Calabi flow extended with bounded curvature integrals.
problem Extending Calabi flow on compact Kähler manifolds.
method Using bounded Lp scalar curvature integrals. result Calabi flow can be extended under certain curvature conditions.
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
New bounds on scalar curvature for metric sequences.
problem Bounding scalar curvature in metric sequences.
method Integral convergence of scalar curvature; point-wise scalar curvature lower bound.
result Limiting metric has scalar curvature lower bound.
The paper shows inequality and rigidity for manifolds with integral Ricci curvature.
problem Analyzing structures of manifolds with integral Ricci curvature.
method Using segment inequality and similar methods as in \cite{CC1}, derive almost rigidity structure results.
result Sharp Hölder continuity result holds in the limit space of manifolds with integral Ricci curvature bound.
Willmore-type inequalities for bounded domains in manifolds with curvature bounds.
problem Establishing inequalities for bounded domains in manifolds with curvature bounds.
method Using asymptotic or integral Ricci curvature bounds to establish inequalities.
result Recovering a recent inequality of Jin-Yin.
Upper bounds on Betti numbers via curvature constraints.
problem Bounding Betti numbers of Riemannian manifolds.
method Integral bounds on curvature eigenvalues, Bochner technique.
result New curvature condition for vanishing Betti numbers.
In this note we give a heat kernel lower bound in term of integral Ricci curvature, extending Cheeger-Yau's estimate.
We consider solutions (M,g(t)), 0 <= t <T, to Ricci flow on compact, four dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case that the scalar curvature is bounded and T is finite, we show that these estimates imply that the (spatial) integral …
Example surfaces with curvature bounds but no Laplacian eigenvalue lower bound.
problem Bounding curvature does not guarantee positive Laplacian eigenvalues.
method Explicit construction of surfaces with bounded curvature and diameter.
result Found surfaces without positive Laplacian eigenvalue lower bound.
Constructs flows on manifolds with small curvature, proving Euclidean topology.
problem Geometric structure of manifolds with unbounded curvature.
method Distance like functions with integral hessian bound, Ricci flows.
result Manifolds with Ricci lower bound, non-negative scalar curvature, bounded entropy, Ahlfors n-regular and small curvature concentration are topologically Euclidean. The paper improves estimates on singular sets in manifolds with integral curvature bounds.
problem Estimating the singular set of manifolds with integral curvature bounds.
method Using Gromov-Hausdorff limits and Cheeger-Naber methods.
result Improved Minkowski dimension estimate for singular sets.
Paper bounds the lowest spectrum of manifolds with curvature constraints.
problem Estimating the lowest energy level of manifolds with curvature restrictions.
method Used a minimal positive Green's function and its properties.
result Proved an upper bound for the bottom of the spectrum.
Estimates the mass gap for domains with integral Ricci curvature bounds.
problem Estimating the mass gap for domains with specific curvature conditions.
method Proving domains are John domains to estimate the first nonzero Neumann eigenvalue.
result Fundamental gap estimates for domains with integral Ricci curvature bounds.
In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.
Integral of scalar curvature over a manifold is bounded by a constant depending on dimension and curvature threshold.
problem Bounding curvature integral over open manifolds and smooth manifolds with boundary.
method Induced Riemannian metric on tangent cones.
result Integral of scalar curvature over a smooth manifold is bounded.
The paper studies Ricci flow with finite curvature integrals on manifolds.
problem Finite curvature integrals on closed manifolds.
method Ricci flow with integral curvature bounds.
result The flow converges to a smooth manifold except for orbifold singularities.
The paper proves a quantitative rigidity result for spaces with specific curvature bounds.
problem Understanding the rigidity of spaces with almost maximal volume entropy.
method Analyzing Riemannian manifolds and RCD-spaces with specific curvature conditions. result Spaces with almost maximal volume entropy are closely related to hyperbolic space forms.
Along a Ricci flow solution on a closed manifold, we show that if Ricci curvature is uniformly bounded from below, then a scalar curvature integral bound is enough to extend flow. Moreover, this integral bound condition is optimal in some sense.
We prove a pseudolocality type theorem for compact Ricci Flow under local integral bounds of curvature. The main tool is Local Ricci Flow introduced by Deane Yang in [4] and Pseudolocality Theorem of Perelman in [3]. We also study L^p bounds for the derivatives of curvature and smooth extension of Local Ricci Flow.
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>Λ1(Sn,[gst.]).
We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the L2 Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.
Proves Ricci flow extensibility with integral norms.
problem Ricci flow singularities under integral norms.
method Bounded integral norms of curvature and scalar curvature.
result Extends Ricci flow under certain conditions.
In this paper, we study the integral curvatures of Finsler manifolds and prove several Myers type theorems.
Study extends compactness theorems to weighted manifolds with integral curvature bounds.
problem Estimating diameter of weighted manifolds under curvature constraints.
method Extended Sprouse's compactness theorems to weighted manifolds with integral curvature bounds. Used ε-range to handle specific cases. Extended segment inequality to weighted manifolds.
result Proved theorems for weighted manifolds with effective dimension ≤ 1 and ≥ dimension.
This article shows that under locally uniformly integral bounds of the negative part of Ricci curvature the heat kernel admits a Gaussian upper bound for small times. This provides general assumptions on the geometry of a manifold such that certain function spaces are in the Kato class. Additionally, the results imply …
The paper derives new gradient estimates for nonlinear elliptic equations under integral Ricci curvature bounds.
problem Gradient estimates for nonlinear elliptic equations under integral Ricci curvature bounds.
method Moser's iteration method applied to positive solutions.
result New local and global gradient estimates for positive solutions are derived.
New connections found between curvature and Euler characteristic using Schrödinger operators.
problem Establishing relationships between curvature and topological invariants of Riemannian manifolds.
method Using twisted Dirac operators and scaling of potentials to analyze the kernel of these operators.
result Found conditions under which the Euler characteristic of a manifold can be zero or non-zero.
We prove a sharp Zhong-Yang type eigenvalue lower bound for closed Riemannian manifolds with control on integral Ricci curvature.
We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface Σ. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation of singularities is allowed.
Study compares nonsmooth spaces with integrable Ricci bounds.
problem Comparing geometric and functional inequalities on nonsmooth spaces.
method Localization method and one-dimensional comparison estimates.
result Extension of comparison principles to nonsmooth settings.
Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.
problem Proving long-time existence and topological rigidity for manifolds with pinched scale-invariant integral curvature.
method Proves long-time existence of Ricci flow for manifolds with bounded curvature and pinched scale-invariant integral curvature, converging to a flat metric.
result Flow converges to a flat metric, implying topological rigidity of the manifold.
Paper compares total quotient curvature and proves bounds for Einstein metric.
problem Comparing total quotient curvature and proving bounds for Einstein metric.
method Established comparison theorems and proved integral inequalities.
result Background Einstein metric achieves a sharp bound on total quotient curvature.
The paper establishes Harnack inequalities for solutions of nonlinear parabolic equations on manifolds with integral Ricci curvature bounds.
problem Analyzing solutions of nonlinear parabolic equations on manifolds with specific curvature constraints.
method Establishing space-time gradient estimates and integrating them to find Harnack inequalities.
result Harnack inequalities for positive solutions of nonlinear parabolic equations under integral Ricci curvature bounds.
Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
The paper derives inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
problem Geometric inequalities for mean curvatures of hypersurfaces in Riemannian manifolds.
method Comparison formula via Reilly's identities; geometric inequalities derived.
result Sharp lower bound for total first mean curvature in dimension 3.
The paper proves geometric comparisons on metric measure spaces with integral Bakry-Émery Ricci tensor bounds.
problem Geometric comparisons on metric measure spaces with specific tensor bounds.
method Integral radial Bakry-Émery Ricci tensor bounds and potential function/gradient bounds.
result Diameter and eigenvalue estimates on smooth metric measure spaces.
We provide integral curvature bounds for compact Riemannian manifolds that allow isometric immersions into a Euclidean space with low codimension in terms of the Betti numbers.
Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
The study examines stability of metric measure spaces with integral Ricci curvature bounds.
problem Stability and compactness of metric measure spaces with integral Ricci curvature bounds.
method Proves convergence to metric measure spaces satisfying CD(K,n) condition under certain curvature bounds. result Proves convergence of sequences of Riemannian manifolds to metric measure spaces satisfying CD(K,n) condition. Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for p<1/2 we show the existence of bounds on the local Lp norm of the Ricci curvature that depend only on the dimension and which improve with volume collapse.
In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that the integral of the mean curvature is infinity.