New metric defined for bounded symmetric domains.
problem Defining a new metric for bounded symmetric domains.
method Using generalized Hilbert metric and Borel embedding.
result The new metric differs from Carathéodory and Bergman metrics except for complex hyperbolic space.
The paper studies complex Finsler metrics and their equivalence to the Kobayashi metric.
problem Investigating properties and equivalence of complex Finsler metrics.
method Using curvature properties of Bergman metrics and Schwarz lemma, the paper analyzes complex Finsler metrics and their equivalence to the Kobayashi metric.
result Uniform equivalences of the Kobayashi metric and Carathéodory metric on bounded strongly convex domains with smooth boundaries are proven.
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
The study examines metrics on Riemannian spaces with bounded properties and finds conditions for Lipschitz and uniform bounds.
problem Investigating bounded rough Riemannian metrics and their properties.
method Analyzing the structure of bounded rough Riemannian metrics and finding conditions for Lipschitz and uniform bounds.
result Weak conditions are identified for Lipschitz and uniform bounds on the metrics.
Ricci flow stability on manifolds with bounded geometry ensures convergence to hyperbolic metrics.
problem Stability and convergence of Ricci flow on manifolds with bounded geometry.
method Continuous dependence on initial conditions, sectoriality of Ricci-DeTurck flow generator, and Hölder norm analysis.
result Ricci flow converges to hyperbolic metrics under certain conditions.
We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds on the Ricci tensor and a lower injectivity radius bound we obtain a uniform estimate on the change of the sectional curvature. Actually, our result holds for any metric with a uniform bound on the W2,p-harmonic radius…
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.
New bounds for low-regularity Riemannian metrics defined via distributional curvature.
problem Establishing curvature bounds for Riemannian metrics of low regularity.
method Introducing a distributional version of sectional curvature for C1 and C0 metrics. result New bounds for low-regularity metrics recover classical bounds in Alexandrov spaces.
We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen-LeBrun-Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the calculation of the invarian…
Surveying Ricci flow for weak lower scalar curvature bounds.
problem Creating local definitions for weak lower scalar curvature bounds for C0 metrics. method Using Ricci flow to define and analyze weak lower scalar curvature bounds.
result Properties and applications of Ricci flow in defining weak lower scalar curvature bounds.
Theorem proves spectral rigidity of warped product metrics.
problem Spectral rigidity of warped product metrics.
method Spinor and spacetime harmonic function methods.
result Proves spectral Llarull theorem for warped product metrics.
In this paper, we consider a CscK metric defined away from divisor and with metric upper bound and lower bound going to zero in certain rate. And we'll prove that this "nicely" behaved metric is a smooth CscK metric across the divisor.
A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the initial metric and which converges to the profile of the constant curvature metric. Th…
Based on C. Li and Y. Rubinstein's upper bisectional curvature bound estimate for the conic Kähler metric, we can construct a smoothing sequence for the conic metric with uniformly upper bisectional curvature bound. For the conic metric along a simple normal crossing divisor with triple or higher multiple points we may…
This note relaxes conditions for Kähler metrics with bounded entropy and scalar curvature.
problem Boundedness conditions for Kähler metrics with bounded entropy and scalar curvature.
method Slightly relaxes the boundedness condition on the scalar curvature.
result Apriori estimates and C3,α estimate for the potential of the Kähler metrics under relaxed conditions. The paper provides generalization bounds for metric learning using neural network embeddings.
problem Generalization guarantees for metric learning with neural network embeddings.
method Uniform generalization bounds for two regimes: sparse and bounded amplification.
result Dimension-free generalization bounds can be achieved even without sparsity in solutions.
The paper establishes lower bounds on injectivity radius and constructs metrics with bounded geometry.
problem Establishing lower bounds on the normal injectivity radius of hypersurfaces and constructing metrics with bounded geometry on manifolds with boundary.
method Pointwise lower estimates and constructions of metrics with bounded geometry.
result The construction of metrics with bounded geometry on arbitrary manifolds with boundary.
Study of Lagrangian submanifolds with Riemannian bounds and their metric properties.
problem Understanding the geometry and topology of Lagrangian submanifolds with Riemannian constraints.
method Investigation of metric properties and symplectic structures on spaces of Lagrangian submanifolds with uniform Riemannian bounds.
result There are at most countably many Hamiltonian isotopy classes of exact Lagrangian submanifolds in a Liouville manifold.
We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds.
In this paper we continue our study on the canonical metrics on the Teichmüller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carathéodory metric to the Kähler-Einstein metric, solving another old conjecture of Yau. We then prove that the Ricci curvature of the p…
Sharp upper bounds on inscribed radius for metric spaces with convex boundary.
problem Bounding inscribed radius in metric measure spaces with convex boundary.
method Proves sharp upper bounds on inscribed radius for subsets with convex boundary.
result Sharp upper bounds on inscribed radius for subsets with convex boundary.
New bounds use IPMs to improve generalization in machine learning.
problem Improving generalization bounds in machine learning.
method PAC-Bayes bounds with Integral Probability Metrics (IPM).
result Natural interpolation between worst-case and favorable cases.
3D metrics get scalar curvature bounds via IMCF.
problem Bounding scalar curvature for C0 metrics. method Inverse Mean Curvature Flow (IMCF) and Hawking mass monotonicity.
result Stability theorem for nonnegative scalar curvature.
The study proves the existence of complete Kähler metrics with negative holomorphic bisectional curvature in specific domains.
problem Proving the existence of complete Kähler metrics with negative holomorphic bisectional curvature in certain domains.
method Analyzing bounded domains in Cn with specific curvature properties. result Strictly pseudoconvex bounded domains and domains with squeezing function tending to 1 at boundary points admit complete Kähler metrics with negative holomorphic bisectional curvature everywhere.
The paper describes invariant twisted Kähler-Einstein metrics on flag varieties.
problem Existence and properties of invariant twisted Kähler-Einstein metrics on flag varieties.
method Invariant twisted Kähler-Einstein metrics on flag varieties, exploring applications and inequalities.
result Established inequalities related to optimal volume upper bounds for Kähler metrics.
We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved S1-invariant metrics on CP1 to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metri…
Study bounds on curvature for special Finsler metrics.
problem Curvature and topological properties of ∞-Einstein Finsler metrics. method Construct special metrics, analyze equivalence, impose curvature bounds.
result Establish bounds for curvature and distortion on ∞-Einstein Finsler manifolds. We apply Tian's method in Kahler-Einstein problem to prove that a conic K\''ahler metric with lower Ricci curvature bound can be approximated by smooth K\''ahler metrics with the same lower Ricci curvature bound. Furthermore, conic singularities here can be along a simple normal crossing divisor.
Bounds on geodesic distances on Stiefel manifold derived from new metrics.
problem Improving geodesic computation algorithms and understanding Stiefel manifold.
method New geometric insights and Lipschitz constants for geodesic distances.
result Explicit bounds on geodesic distances and conditions for attaining bounds.
We consider the moduli space of the extremal Kähler metrics on compact manifolds. We show that under the conditions of two-sided total volume bounds, L2n-norm bounds on $\Riem$, and Sobolev constant bounds, this Moduli space can be compactified by including (reduced) orbifolds with finitely many singularities…
Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.
problem Understanding the relationship between intrinsic and extrinsic metrics and surface area.
method Examined surfaces within a unit ball in R3, provided lower bounds on the ratio in terms of area, and showed non-existence of global lower bounds.
result Found that the ratio of intrinsic to extrinsic metrics has a lower bound in terms of surface area, but no global lower bound exists.
Study compares synthetic and distributional Ricci curvature bounds.
problem Comparing synthetic and distributional approaches to lower Ricci curvature bounds.
method Analyzes synthetic via weak displacement convexity and distributional via non-negativity of Ricci-tensor.
result Distributional bounds imply entropy bounds for C1 metrics and vice versa for C1,1 under convergence condition. Study on spaces of metrics with intermediate curvature bounds.
problem Understanding spaces of metrics with lower bounds on intermediate curvatures.
method Analyzing spaces of Riemannian metrics with specific curvature bounds on high-dimensional Spin-manifolds.
result Spaces of metrics with positive p-curvature and k-positive Ricci curvature have non-trivial homotopy groups.
Survey on warped products and their curvature properties.
problem Understanding warped products and their curvature bounds.
method Construction and analysis of warped products between manifolds and metric spaces.
result Warp products have nice curvature properties, especially sectional and Ricci bounds.
This note is a continuation of the author's paper \cite{Li}. We prove that if the metric g of a 4-manifold has bounded Ricci curvature and the curvature has no local concentration everywhere, then it can be smoothed to a metric with bounded sectional curvature. Here we don't assume the bound for local Sobolev constan…
Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.
problem Bounding Laplacian eigenvalues on manifolds with non-negative scalar curvature.
method Investigation of invariant spectrum on compact Riemannian manifolds with large isometry groups.
result Upper bounds for eigenvalues of the invariant spectrum assuming non-negative scalar curvature.
We first give a precise statement on the short time existence of the Calabi flow and prove a stability result: any metric near a constant scalar curvature metric will flow to this cscK metric exponentially fast. Secondly, we prove that a compactness theorem in space of the kahler metrics given unifrom Ricci bound and p…
We prove the equivalences of several classical complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new Kähler metrics, the Ricci metric and the perturbed Ricci metric and prove that the perturbed Ricci metric is a complete Kähler metric with bounded negative holomorphic s…
Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.
problem Eigenvalue problems for Bi-drifted Laplacian on compact manifolds with boundary conditions.
method Obtained lower bounds using specific curvature conditions.
result Lower bounds for the first eigenvalue of Bi-drifted Laplacian.
Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.
problem Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
method Established optimal upper bounds for cone angles of Kähler-Einstein metrics with conical singularities.
result Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
Sharp bounds on neural network approximation rates and widths.
problem Estimating approximation rates, metric entropy, and n-widths of shallow neural networks.
method Introducing smoothly parameterized dictionaries and providing upper and lower bounds.
result Sharp bounds on approximation rates, metric entropy, and n-widths for neural networks with various activation functions.
On any surface we give an example of a metric that contains simple closed geodesics with arbitrary high Morse index. Similarly, on any 3-manifold we give an example of a metric that contains embedded minimal tori with arbitrary high Morse index. Previously no such examples were known. We also discuss whether or not suc…
We produce complete bounded curvature solutions to Kähler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth \K metric uniformly equivalent to another complete bounded curvature \K metric. We obtain related flow results for non-smooth as well as degenerate initial conditions. We a…
Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.
problem Smooth metrics on manifolds with curvature and injectivity constraints.
method Bi-Lipschitz smoothing with controlled smoothing and volume lower bounds.
result Proves existence of smooth metrics with curvature bounds and injectivity radius constraints.
The norm of Cartan torsion plays an important role for studying of immersion theory in Finsler geometry. Indeed, Finsler manifold with unbounded Cartan torsion can not be isometrically imbedded into any Minkowski space. In this paper, we find two subclasses of (?, ?)-metrics which have bounded Cartan torsion. Then, we …
Sharp eigenvalue bounds on metric measure spaces extend Cheng's theorem.
problem Extending Cheng's eigenvalue comparison theorem to non-smooth spaces.
method Localization technique, synthetic Ricci curvature bounds via optimal transport.
result Sharp upper bounds on eigenvalues in metric measure spaces.
We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…
Uniform Sobolev inequality for Kähler metrics with entropy bound.
problem Establishing Sobolev inequalities for Kähler metrics with entropy bound.
method Uniform Sobolev inequality for Kähler metrics with entropy bound and no lower Ricci curvature bound.
result Derive various geometric estimates for Kähler-Einstein currents.