The anomaly flow on a complex 3-fold is studied with integral Shi-type estimates and long-time existence conditions.
problem Long-time existence of the anomaly flow on a compact complex 3-fold.
method Integral Shi-type estimates adapted from integration-by-parts arguments, with a smallness condition on the slope parameter.
result Long-time existence of the anomaly flow on a compact complex 3-fold under a smallness condition on the slope parameter.
In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to more general flows of G2 structures including the modified Laplacian co-flow. Then we prove a version of κ-non-collapsing theorem. We will use both of them to study finite time singularities of general flows of G2 structures.
The paper analyzes finite-time singularities in Spin(7)-structure flows using Shi-type estimates.
problem Analyzing finite-time singularities in Spin(7)-structure flows.
method Proves Shi-type derivative estimates and shows that Λ(x,t) must blow up at finite-time singularities.
result Establishes a general analytic framework for studying Spin(7)-structure flows.
We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all n-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with ∣Ric∣≤K must satisfy $|Rm|\…
The paper provides gradient estimates for a parabolic equation under Finsler geometric flows.
problem Gradient estimates for a general parabolic equation under compact Finsler CD(−K,N) geometric flows. method Presented Shi-type and Hamilton-type gradient estimates.
result Demonstrates the possibility of removing stricter derivative bounds imposed by Finsler curvature conditions.
In this note we obtain local derivative estimates of Shi-type for the heat equation coupled to the Ricci flow. As applications, in part combining with Kuang's work, we extend some results of Zhang and Bamler-Zhang including distance distortion estimates and a backward pseudolocality theorem for Ricci flow on compact ma…
Geometric flow on symplectic manifolds connects to Type IIA string theory.
problem Finding optimal almost-complex structures compatible with symplectic forms.
method Introduced a geometric flow on 6-dimensional symplectic manifolds with SU(3) holonomy.
result The flow leads to Ricci-flat Kähler metrics and optimal almost-complex structures.
We develop foundational theory for the Laplacian flow for closed G_2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on $Λ(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\ri…
Study harmonic flow of Spin(7)-structures on compact 8-manifolds.
problem Isometric flow of Spin(7)-structures on compact 8-manifolds.
method Establishing Shi-type estimates, self-similar solutions, monotonicity formula, compactness theorems, and Bryant-type description.
result Conditions for long-time existence and characterisation of singularities.
We survey recent progress in the study of G2-structure Laplacian coflows, that is, heat flows of co-closed G2-structures. We introduce the properties of the original Laplacian coflow of G2-structures as well as the modified coflow, reviewing short-time existence and uniqueness results for the modified co…
Ansatze are constructed under which the solutions of 11D supergravity must be stationary points of a parabolic flow on a Riemannian manifold M10−p. This parabolic flow turns out to be the Ricci flow coupled to a scalar field, a (3−p)-form, and a 4-form. This allows the introduction of techniques from parabolic…
New flow for G2-structures helps find torsion-free structures.
problem Finding torsion-free G2-structures on compact manifolds.
method Ricci-harmonic flow of G2-structures, analyzing Taylor series expansion.
result Stationary points of the flow are torsion-free G2-structures.
Real analyticity proved for modified Laplacian coflow solutions.
problem Analyzing the modified Laplacian coflow on compact 7-manifolds.
method Improved Chen's Shi-type estimate to show real analyticity.
result The modified Laplacian coflow solution is real analytic.
Flow solves G2 system, proving existence of torsion-free metrics.
problem Existence of large volume heterotic G2 solutions. method Geometric flow of conformally coclosed G2-structures. result Fundamental short-time existence and smoothing properties established.
We study a flow of G2 structures which induce the same Riemannian metric which is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor along the flow. We show that at a finite-time singularity the torsion must blow-up, so the flow exists as long as the torsion remain…
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.
Gradient and Laplacian estimates for complex Monge-Ampère equations found.
problem Estimating solutions to complex Monge-Ampère equations with singularities.
method Integral method applied to obtain gradient and Laplacian estimates.
result Gradient and Laplacian estimates for the solution to the singular complex Monge-Ampère equation.
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. 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.
Improved estimation of higher order integrals using shrinkage techniques.
problem Estimating higher order Bochner integrals in non-parametric settings.
method Shrinkage of U-statistic towards a target element, considering kernel degeneracy.
result Consistent shrinkage estimators with fast rates of convergence, even for non-degenerate kernels.
The paper uses Fourier integral theorem for estimating multivariate distributions.
problem Estimating multivariate distributions and conditional distribution functions.
method Natural Monte Carlo and fully nonparametric estimators based on Fourier integral theorem.
result Explicit Monte Carlo estimators without estimated covariance matrix.
New integral theorems improve density function estimations.
problem Improving density function estimations.
method Integrals based on cyclic functions and Riemann sums, Fourier integral theorem, Monte Carlo methods, variational approach, Cauchy residue theorem.
result Optimal cyclic functions minimize square integrals, improving density estimations.
We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry-Émery Ricci tensor bounds, extending Wei-Wylie's comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates and volume growth estimates on smoo…
Paper extends curvature estimates to new tensor types.
problem Mean curvature and volume comparison estimates for integral generalized quasi-Einstein tensors.
method Extends existing comparison results to new tensor types.
result Global diameter estimates derived from comparison results.
The paper provides gradient estimates for solutions on manifolds with integral Ricci bounds.
problem Global regularity estimates for solutions of Δu=f on Riemannian manifolds. method Proves Lp-gradient estimates under integral Ricci bounds and constructs a counterexample. result Optimal constant lower bounds on Ricci curvature are shown in the pointwise sense.
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.
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.
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.
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 integral estimates on substatic manifolds improve Alexandrov Theorem.
problem Improving integral estimates on substatic manifolds.
method Introducing a new vector field with nonnegative divergence.
result Generalization and improvement of integral estimates leading to Alexandrov Theorem.
Study critical metrics on manifolds with boundary using integral and boundary estimates.
problem Investigate geometry of critical metrics on compact manifolds with boundary.
method Use generalized Reilly's formula to derive integral and boundary estimates.
result Establish new boundary estimates for critical metrics of the volume functional.
BMTI method estimates densities without bins, outperforming traditional estimators.
problem Nonparametric, robust, and data-efficient density estimation in high-dimensional spaces.
method BMTI integrates log-density differences between neighboring points, weighted by uncertainties, using a maximum-likelihood formulation.
result BMTI reconstructs smooth profiles in high-dimensional spaces, outperforming traditional estimators.
Paper provides estimates for varifolds with critical mean curvature.
problem Estimating tilt-excess on varifolds with critical mean curvature.
method Generalizing Lipschitz approximation and Sobolev-Poincaré estimates to almost-integral rectifiable varifolds.
result VMO-type estimates for quadratic tilt-excess on varifolds with critical mean curvature.
Study extends convexity in curved spaces using fractional integrals.
problem Extending convexity to curved spaces with nonpositive curvature.
method Introducing (geodesically) h-convex functions and using Katugampola's fractional integrals. result Essentially sharp estimate involving squared distance mappings.
We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local bounds. In doing so, we prove for the first time that the Lie-Butcher theory of Lie group integrators leads to global error estimates.
The paper estimates area and volume for spacetimes with integral mean curvature bounds.
problem Estimating area and volume for spacetimes with specific curvature conditions.
method Using strong energy condition and norms of second fundamental form/mean curvature.
result Established area and volume estimates for spacetimes.
The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
problem Gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
method Assumes a Sobolev inequality and integral Ricci bounds, proving local gradient estimates and Liouville type results.
result Proves local gradient estimates and Liouville type results on manifolds with lower bounds of Ricci curvature.
Q-NETs use neural networks to estimate integrals of low-dimensional functions efficiently.
problem Estimating integrals of multidimensional functions with costly evaluations.
method Fixed neural networks (Q-NETs) that operate on proxy function parameters to calculate exact integrals over subsets of dimensions.
result Q-NETs can calculate integrals over any subset of dimensions without resampling or retraining the proxy.
Monte Carlo (MC) techniques are often used to estimate integrals of a multivariate function using randomly generated samples of the function. In light of the increasing interest in uncertainty quantification and robust design applications in aerospace engineering, the calculation of expected values of such functions (e…
Two EP frameworks ensure integrable beliefs in Bayesian estimation problems.
problem Non-integrable beliefs in EP can lead to infeasible solutions.
method Proposes two EP frameworks to keep messages non-integrable.
result Ensures integrable beliefs in EP, even with non-integrable messages.
The paper introduces new estimators for multivariate functions using Fourier methods.
problem Estimating multivariate functions like densities and regression functions.
method Monte Carlo estimators based on the Fourier integral theorem.
result Established rates of convergence for new estimators, often superior to existing methods.
Estimates submanifold diameters in curved spaces.
problem Estimating the intrinsic diameter of submanifolds in curved spaces.
method Using mean curvature field integrals and boundary lengths.
result Diameter estimates for submanifolds in curved spaces.
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 …
Proof shows volume equals integral points for certain manifolds.
problem Counting integral points on affine manifolds.
method Rational Ehrhart theory and Fourier analysis.
result Volume equals number of integral points for integral-integral affine manifolds.
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.
We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in \cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate in \cite{Au2} and giv…
Estimates gradients of solutions on closed surfaces.
problem Gradient estimates for solutions on closed surfaces.
method Considered a new metric g′=e2ug with bounded integral curvature, derived gradient estimates for g′, and used these to obtain gradient estimates for u. result Gradient estimates for solutions on closed surfaces are established.
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.