Study finds rigidity of biconservative hypersurfaces in space forms without curvature assumptions.
problem Investigating biconservative hypersurfaces in space forms without scalar curvature assumptions.
method Introduced a novel divergence-free tensor to derive results without curvature assumptions.
result Rigidity results for biconservative hypersurfaces in space forms without scalar curvature assumptions.
We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…
Derives formulas from Green function Hessian assumption.
problem Deriving formulas from Green function Hessian assumption.
method Assumption on Hessian of Green function leads to monotonicity formulas.
result Explicit examples of manifolds satisfying assumption.
In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant sectional curvature is replaced by the assumption that the curvature satisfies some inequalities.
New curvature assumptions prove Nakano positivity for complex vector bundles.
problem Proving Nakano positivity for complex vector bundles under varying curvature assumptions.
method Using a variant of Hörmander's theorem, the authors show Nakano positivity under more general curvature conditions.
result Nakano positivity holds for complex vector bundles under different curvature assumptions.
Study improves understanding of Ricci curvature in manifolds.
problem Understanding Ricci curvature in manifolds with specific assumptions.
method Exploring m-intermediate Ricci curvature and proving comparison theorems.
result Stable weighted slicing in manifolds with non-negative m-intermediate Ricci curvature has almost non-negative Ricci curvature.
Study on evolving graphs of functions under mean curvature flow in R^n.
problem Proving long-time existence and convergence of special Lagrangian evolution equation.
method Consider the graph of a C2 function u on Rn, deform it by mean curvature flow, and analyze under 2-positivity assumption. result Proves long-time existence and convergence results under 2-positivity assumption, improving previous results.
We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature assumption and no assumption on sign of the sectional curvature…
Paper finds conditions for minimal hypersurfaces in S^6 with constant scalar curvature.
problem Finding conditions for minimal hypersurfaces in S^6 with constant scalar curvature.
method Assumptions on principal curvatures for isoparametric hypersurfaces.
result Rigidity result: Hypersurfaces with exactly two distinct principal curvatures are Clifford tori.
New method removes scalar curvature assumption in Ricci flow smoothing.
problem Uniform bounds on scalar curvature and other factors for Ricci flow.
method Quantitative short-time existence of Ricci flow without scalar curvature assumption.
result Ricci flow smoothing for measure space limits, Gromov-Hausdorff compactness, and topological rigidity results.
Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow preserves the area enclosed by the evolving curve, and this fact is used to prove a gene…
Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci…
The paper improves eigenvalue estimates for manifolds with Ricci curvature conditions.
problem Eigenvalue estimates for manifolds with Ricci curvature conditions.
method Proves eigenvalue estimates using a Kato condition on the negative part of Ricci curvature.
result Optimal eigenvalue estimates for Zhong-Yang type and Cheng-type bounds.
We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold (Mn,g) for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schrödinger operator (n−2)Δ+ρ is positive has finite fundamental group. As a continuation of our earlier results, we obtain isoperimetric inequa…
Study diameter bounds on Kähler and quaternionic Kähler manifolds with positive curvature.
problem Determine diameter bounds for Kähler and quaternionic Kähler manifolds under curvature positivity.
method Define orthogonal Bakry-Émery tensor, study diameter theorems, and derive Bonnet-Myers type bounds.
result Sharper diameter bounds than in Riemannian case under specific curvature assumptions.
The paper studies curvature conditions on manifolds with boundary.
problem Curvature preservation on manifolds with smooth boundaries.
method Constructing a family of metrics that agree with given metrics on the boundary and interior.
result Deforming metrics to ones with totally geodesic boundary while preserving curvature conditions.
We prove curvature-free versions of the celebrated Margulis Lemma. We are interested by both the algebraic aspects and the geometric ones, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of some important invariants. Our goal is to get rid of the pointwise curvature assum…
In [8] Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Further…
Prove that collapsing CSC metrics can be perturbed to invariant collapsing CSC metrics.
problem Prove that collapsing constant scalar curvature metrics can be perturbed to invariant collapsing constant scalar curvature metrics.
method Prove that a sequence of constant scalar curvature metrics which is collapsing with bounded curvature to a manifold can be perturbed to a sequence of invariant collapsing constant scalar curvature metrics.
result Prove that a sequence of constant scalar curvature metrics which is collapsing with bounded curvature to a manifold can be perturbed to a sequence of invariant collapsing constant scalar curvature metrics.
Paper relaxes convexity assumptions in mean curvature flow results.
problem Relaxing convexity assumptions in mean curvature flow results.
method Proves a generalized Harnack inequality and uses maximum principle.
result Characterizes family of shrinking spheres for ancient solutions.
Improved heat equation estimates without gradient curvature assumption.
problem Improving Hamilton's matrix Harnack estimate for heat equation without gradient curvature assumption.
method New ingredients include a sharp Li-Yau estimate, a suitable vector field construction, and integral arguments.
result Removed the gradient curvature assumption in Hamilton's estimate for heat equation.
Sampling random points can reveal submanifold topology.
problem Estimating the topology of submanifolds in Riemannian manifolds.
method Sampling random points in a neighborhood of the submanifold.
result Topology of the submanifold can be recovered with high confidence.
We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, …
In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient mani…
We consider open globally hyperbolic spacetimes N of dimension n+1, n≥3, which are spatially asymptotic to a Robertson-Walker spacetime or an open Friedmann universe with spatial curvature κ~=0,−1 and prove, under reasonable assumptions, that there exists a unique foliation by hypersurfaces of constant…
We study convex entire graphs evolving with normal velocity equal to a positive power of the mean curvature. Under mild assumptions we prove longtime existence.
In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of Rn+1 (n⩾2) is mean convex and star-shaped. Several interesting examples and some hyperbol…
Generalizes Ricci flow starting from small curvature concentration with a Morrey-type condition.
problem Ricci flow starting from manifolds with unbounded curvature.
method Replaces bounded curvature with a Morrey-type condition on the gradient of the metric relative to a complete bounded curvature metric.
result Long-time existence of Ricci flow with curvature decay estimates and diffeomorphic manifold.
We study curvature pinching estimates of Ricci flow on complete 3- dimensional manifolds without bounded curvature assumption. We will derive some general curvature conditions which are preserved on any complete solution of 3-dim Ricci flow, these conditions include nonnegative Ricci curvature and sectional curvature a…
In this article, we prove gradient estimates under Bakry-Emery curvature bounds for unbounded graph Laplacians which satisfy an ellipticity assumption. As applications, we study completeness and finiteness of stochastically complete graphs under Bakry-Emery curvature bounds.
On an asymptotically flat manifold Mn with nonnegative scalar curvature, with outer minimizing boundary Σ, we prove a Penrose-like inequality in dimensions n<8, under suitable assumptions on the mean curvature and the scalar curvature of Σ.
For a proper action by a locally compact group G on a manifold M with a G-equivariant Spin-structure, we obtain obstructions to the existence of complete G-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where M/G is noncompact. The obstructions follow from a Callia…
The paper improves bounds on injectivity radius for manifolds with positive scalar curvature.
problem Finding tighter bounds on the injectivity radius for manifolds with positive scalar curvature.
method Utilizing Green's inequality and topological assumptions on manifolds, including specific 3-manifolds and products.
result Stronger upper bounds on injectivity radius for certain manifolds, including products and 3-manifolds with positive scalar curvature.
In this survey article we will consider universal lower bounds on the volume of a Riemannian manifold, given in terms of the volume of lower dimensional objects (primarily the lengths of geodesics). By `universal' we mean without curvature assumptions. The restriction to results with no (or only minimal) curvature assu…
Study geometric structure of Ricci shrinker ends without global curvature assumptions.
problem Understand the geometric structure of Ricci shrinker ends without global curvature constraints.
method Analyze blow-up sequences of Ricci shrinkers at points with Type I scalar curvature bound, extending F-convergence theory.
result Limits of Ricci shrinkers at points with Type I scalar curvature bound split a line in four dimensions.
Ancient Lagrangian flows get limited convex solutions.
problem Controlling convex solutions of Lagrangian flows at antiquity.
method Proving a Liouville type theorem with quadratic growth restrictions.
result Ancient convex solutions are unique.
Proves existence of proper solutions for inverse mean curvature flow.
problem Existence of proper solutions for inverse mean curvature flow.
method Proves existence theorem assuming non-degeneracy conditions on isoperimetric profile.
result No curvature assumption in existence theorem.
The paper generalizes rigidity results for contact Anosov flows with bunching assumption.
problem Rigidity of contact Anosov flows in higher dimensions.
method Application of matching functions technique with bunching assumption.
result If two contact Anosov flows are C0 conjugate, they are Cr conjugate for some r∈[1,2) or even C∞ conjugate under additional assumptions. We obtain a priori estimates for solutions to the prescribed scalar curvature equation on S3. The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function is a positive Morse functi…
Given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero and the evolution equation by T-curvature at the boundary with the condition that the Q-curvature …
New findings show functional inequalities fail on Finsler manifolds with positive S-curvature.
problem Failure of functional inequalities on Finsler manifolds with positive S-curvature.
method Analysis of Finsler metric measure manifolds with reversibility, flag curvature, and S-curvature.
result Functional inequalities fail on Finsler manifolds with positive S-curvature.
Simplified Obata-Vétois argument for Einstein manifolds with nonnegative scalar curvature.
problem Identifying conditions for closed conformally Einstein manifolds to be Einstein.
method Simplified Obata-Vétois argument, identifying a closed interval containing zero.
result Closed conformally Einstein manifolds with nonnegative scalar curvature are Einstein if they satisfy certain conditions.
Study of spacetimes in cosmology without symmetry assumptions.
problem Proving timelike incompleteness and properties of time functions.
method Rigorous mathematical proofs and analysis of spacetime properties.
result Timelike incompleteness for specific spacetimes with mean curvature constraints.
Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.…
Ricci flow controls curvature on manifolds with bounds.
problem Controlling curvature on manifolds with given bounds.
method Ricci flow with curvature bounds and entropy controls.
result Global curvature control at positive times for manifolds.
A convex surface contracting by a strictly monotone, homogeneous degree one function of curvature remains smooth until it contracts to a point in finite time, and is asymptotically spherical in shape. No assumptions are made on the concavity of the speed as a function of principal curvatures.
The paper finds closed hypersurfaces with prescribed mean curvature in non-compact manifolds.
problem Finding closed hypersurfaces with prescribed mean curvature in non-compact manifolds.
method One-parameter prescribed mean curvature min-max theory.
result Closed hypersurfaces with prescribed mean curvature are found in certain non-compact manifolds.
Study classifies harmonic vector fields on 3-manifolds.
problem Classifying harmonic unit vector fields on 3-manifolds.
method Investigates under mild curvature assumptions, classifying vector fields and manifolds.
result Classifies both vector fields and manifolds supporting them.