A new deep learning method using differential flows and Gaussian processes.
problem Improving deep learning performance through more flexible transformations.
method Differential flows and Gaussian processes to learn stochastic differential equations.
result Exceeds performance of deep Gaussian processes and neural networks.
Study links Hopf differentials to curvature line flows on time-like CMC surfaces.
problem Understanding the relationship between Hopf differentials and curvature line flows on time-like CMC surfaces.
method Investigation of Hopf differentials and curvature line flows on time-like CMC surfaces in Lorentzian 3-space forms.
result The index of a curvature line flow at an umbilic point depends on the remainder of the Hopf differential's order modulo four.
Simple connection between Harnack inequalities and concavity of arrival time functions.
problem Proving differential Harnack inequalities for various flows.
method Directly proving concavity properties of time-of-arrival functions for a class of flows using a concavity maximum principle.
result Short proof of Hamilton's and Andrews' differential Harnack inequalities.
Paper introduces a differentially private generative model using gradient flow and sliced Wasserstein distance.
problem Protecting privacy in sensitive training data for generative models.
method Gradient flow in the space of probability measures, Gaussian-smoothed Sliced Wasserstein Distance, and numerical scheme for SDE.
result Demonstrates higher-fidelity data generation at low privacy budget compared to existing methods.
We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the ε-Ricci flow on a closed surface. These new Harnac…
Illustrates Ricci flow for a general audience.
problem Explains Ricci flow to non-experts.
method Illustrative introduction without prerequisites.
result Provides a working definition and intuition for Ricci flow.
Proposes differentially private normalizing flows for privacy-preserving density estimation.
problem Privacy concerns in density estimation models when individuals are directly associated with the training data.
method Uses normalizing flow models with explicit differential privacy guarantees.
result Substantially outperforms previous state-of-the-art approaches in privacy-preserving density estimation.
Non-ergodic measures found in horocycle flow on Abelian differentials.
problem Finding non-ergodic measures in the horocycle flow on Abelian differentials.
method Analyzing weak convergence of ergodic measures to non-ergodic invariant measures.
result Existence of points with non-equidistributing horocycle flow orbits.
Paper proves existence of conjugate points on ellipsoids but not on spheres.
problem Existence of conjugate points in incompressible Euler flows.
method Formulated a differential-geometric criterion (M-criterion) and analyzed flows on spheres and ellipsoids.
result Zonal flows on ellipsoids can satisfy M-criterion, while not on spheres.
Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.
problem Understanding magnetic flows on 3D contact sub-Riemannian manifolds.
method Introducing horizontal magnetic flows via closed Rumin differential two-forms and analyzing the lifted sub-Riemannian structure.
result Horizontal magnetic flows can be interpreted as geodesic flows on a suitably lifted structure, which is of Engel type when the magnetic field is non-vanishing.
This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows o…
Quantum stochastic flow computes heat kernel traces for Ricci flat manifolds.
problem Computing heat kernel traces for Ricci flat manifolds.
method Quantum stochastic differential equation (qsde) on Fock space over L2 differential 1-forms, adapted flow construction. result Trace of the connection Laplacian heat kernel can be computed over any compact Ricci-flat Riemannian manifold.
The paper proves gap results for self-shrinkers in r-mean curvature flow.
problem Understanding the gap in properties of self-shrinkers in r-mean curvature flow. method Proving gap results using a modified second fundamental form and a differential operator.
result Proper self-shrinkers are parabolic for a certain second-order differential operator.
In this paper, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the p-Laplace operator along the Ricci flow on closed manifolds. We show that the first p-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature a…
We convert deterministic flow models to stochastic samplers.
problem Deterministic flow models are sensitive to errors and cannot condition on intermediate states.
method Transform ODEs into SDEs with the same marginal distributions.
result Empirically outperforms deterministic samplers and controls generation diversity.
We construct differential invariants that vanish if and only if the geodesic flow of a 2-dimensional metric admits an integral of 3rd degree in momenta with a given Birkhoff-Kolokoltsov 3-codifferential.
New methods prove existence of rotating shapes moving in space.
problem Existence of rotating shapes moving in space.
method Different methods to prove existence based on singular ordinary differential equation.
result Existence of rotationally symmetric translating solutions proven without partial differential equations.
Motivated by Pan-Yang [PY] and Ma-Cheng [MC], we study a general linear nonlocal curvature flow for convex closed plane curves and discuss the short time existence and asymptotic convergence behavior of the flow. Due to the linear structure of the flow, this partial differential equation problem can be resolved using a…
The study confirms essential self-adjointness for certain differential operators on manifolds.
problem Essential self-adjointness of differential operators on closed manifolds.
method Analyzing the Hamiltonian flow of the symbol of differential operators.
result The conjecture that certain differential operators are essentially self-adjoint if their Hamiltonian flow is complete.
This work proposes a model for geodesic distances and flows on manifolds.
problem Geodesic distances and flows on differentiable manifolds.
method Manifold-augmented Eikonal equation solutions.
result Geodesic flow provides globally length-minimizing curves.
The paper classifies shapes of translating solitons for a specific flow.
problem Understanding the shapes of translating solitons in a specific flow.
method Analyzing functions on a unit sphere and solving an ODE.
result Classification of the shapes of translating solitons.
Given a solution of the (backwards) Ricci flow one can construct a so called canonical soliton metric on space-time, introduced by E. Cabezas-Rivas and P. Topping. We observe that for a mean curvature flow within a (backwards) Ricci flow background, the space-time track of the mean curvature flow yields a canonical sol…
We consider the Whitham equations for deformations of hyperelliptic spectral curves, which preserve all periods of a meromorphic differential. If the meromorphic differential has a root at a fixed point of the hyperelliptic involution, then the Whitham flow has a singularity. We prove that the stable and unstable manif…
We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian 2-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We …
Method learns PDE dynamics via evolving latent manifold using Ricci flow.
problem Learning dynamics in time, especially PDEs, with low-dimensional representations.
method Parameterizes latent manifold, simulates Ricci flow physics-informedly, matching manifold quantities.
result Ricci flow facilitates learning for out-of-distribution data and adversarial robustness.
In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M x R^2 has nonnegative isotropic curvature.
This work develops sampling methods for differential privacy using SHK geometry.
problem Approximating sampling for the exponential mechanism in differential privacy.
method Develops perturbation theory for SHK gradient flows and applies to differential privacy.
result Derives time-dependent Pure-DP guarantees and Approximate-DP certificates.
Harmonic maps from complex plane to hyperbolic space constructed using heat flow.
problem Constructing harmonic maps from complex plane to hyperbolic space.
method Heat flow method to construct harmonic maps.
result Harmonic maps are unique once the principal part of their Hopf differential is prescribed.
In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's …
Gradient flow studies Spin(7)-structures on compact 8-manifolds.
problem Formulating and studying the gradient flow of Spin(7)-structures.
method Negative gradient flow of an energy functional of Spin(7)-structures.
result Short-time existence and uniqueness of solutions to the flow.
The Levy Laplacian is studied on manifolds, with heat flow solutions tending to constant functionals over time.
problem Understanding the Levy Laplacian on manifolds and its heat flow behavior.
method Various definitions of the Levy Laplacian are proven equivalent. Heat flows of differential forms are used to construct solutions.
result Solutions of the heat equation with the Levy Laplacian tend to locally constant functionals over time.
Abstract: Extends geometric concepts to generalized tangent bundle and describes flows.
problem Extend classical geometric notions to generalized geometry.
method Develops differential complexes and generalized connections on the generalized tangent bundle.
result Describes geometric flows and their analogues in generalized geometry.
In this paper we prove first order differential Harnack estimates for positive solutions of the heat equation (in the sense of distributions) under closed Finsler-Ricci flows. We assume mild non-linearities (in terms of the Chern connection, S−curvature and Hessian) and suitable Ricci curvature bounds throug…
In this paper we study the heat equation (of Hodge-Laplacian) deformation of (p,p)-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a (p,p)-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Ya…
Neural ODEs extended to manifolds for flexible sampling.
problem Sampling from complex multimodal distributions on non-trivial topologies.
method Extending Neural ODEs to smooth manifolds using vector fields.
result A general methodology for building normalizing flows on manifolds.
Survey on mean curvature flow with sphere theorems and Yau rigidity theory.
problem Sphere theorems for submanifolds with arbitrary codimension.
method Recent developments on convergence theorems for mean curvature flow.
result Optimal convergence theorem for arbitrary codimension mean curvature flow.
Gradient estimates for a parabolic PDE under Ricci-Bourguignon flow on warped product manifolds.
problem Analyzing the Ricci-Bourguignon flow on warped product manifolds.
method Establishing gradient estimates for a parabolic partial differential equation.
result Gradient estimates for the parabolic PDE provide analytic input for geometric applications.
Study inextensible flows of curves in 4D pseudo-Galilean space and defines energy functions.
problem Analyzing inextensible flows and energy of curves in 4D pseudo-Galilean space.
method Expressed inextensible flows as partial differential equations, defined directional derivatives, and expressed bending elastic energy functions.
result Necessary and sufficient conditions for inextensible flows are given as partial differential equations.
Stochastic normalizing flows use SDEs for efficient training and sampling.
problem Efficient maximum likelihood estimation and variational inference.
method Continuous normalizing flows extended with stochastic differential equations (SDEs) and rough path theory.
result Stochastic normalizing flows enable efficient training and sampling from complex distributions.
New error bounds for flow matching methods using deterministic sampling.
problem Improving the accuracy of flow matching methods for generating probability distributions.
method Derived error bounds for flow matching methods under deterministic sampling conditions.
result Presented error bounds for flow matching methods using L2 loss and regularity conditions. In this paper, we investigate Liu-Xu-Ye-Zhao's conjecture [30] and prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves the convergence theorem of Baker [2] as well as the differentiable sphere theorems of Gu-Xu-Zhao [16, 50, 52].
In this paper, we investigate the general formulation for inextensible flows of curves in En. The necessary and sufficient conditions for inextensible curve flow are expressed as a partial differential equation involving the curvatures.
Study improves Harnack estimates for porous medium equation under geometric flow.
problem Improving Harnack estimates for solutions to the porous medium equation under evolving metrics.
method Differential Harnack estimates for positive solutions to the porous medium equation with potential on time-dependent Riemannian metrics evolving by geometric flow.
result New Harnack estimates for the porous medium equation under geometric flow.
The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the nor…
The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.
problem Understanding ergodicity of geodesic flows on infinite Riemann surfaces.
method Analyzing random walks on the dual graph of pants decompositions.
result Equivalence between ergodicity of geodesic flows and recurrence of random walks.
Study shows swirling flows destabilize boundary regions.
problem Stability of swirling flows near flat boundaries.
method Differential geometric approach to prove instability.
result Proves flows have a destabilizing effect near boundaries.
We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…
The set of directions from a quadratic differential that diverge on average under Teichmuller geodesic flow has Hausdorff dimension exactly equal to one-half.