Eigenvalues of Dirac operator converge on spin manifolds with Riemannian flows.
problem Eigenvalue convergence on manifolds with evolving metrics.
method Riemannian flows and adiabatic limits applied to Dirac operator eigenvalues.
result Eigenvalues converge under Riemannian flow conditions.
Paper introduces entropy for Riemannian foliations and connects it to Ricci flow.
problem Entropy functional for Riemannian foliations and its relation to Ricci flow.
method Introduced entropy functional and related its gradient flow to transverse Ricci flow.
result Entropy functional is monotonic along transverse Ricci flow and related to it.
New geometrical flows derived from Riemannian metrics studied for volume variation and entropy.
problem Volume variation and entropy of Riemannian flows.
method Developed Ricci-Yamabe maps for Riemannian flows and analyzed their volume variation and entropy.
result Derived new examples of Riemannian flows and expressed Ricci flow equations in all four orthogonal separable coordinate systems of the plane.
Riemannian flows model curved spaces better.
problem Normalizing flows misspecified on curved spaces.
method Riemannian continuous normalizing flows using ODE solutions.
result Improves modeling on spheres, torii, and hyperbolic spaces.
Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
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.
In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions…
Study on gauge transformations on Riemannian surfaces with boundary using heat flow method.
problem Analyzing gauge transformations on Riemannian surfaces with boundary.
method Heat flow method for local and long-time existence of solutions.
result Local and long-time existence of generalized solutions proved.
New flow deforms Riemannian metrics smoothly.
problem Deforming Riemannian metrics on spin manifolds.
method Parabolic flow based on Dirac-Einstein functional.
result Local well-posedness of smooth solutions proved.
The paper provides estimates for flows on Riemannian manifolds using truncated expansions.
problem Quantifying the relationship between flows on Riemannian manifolds and their truncated logarithms.
method Using truncated versions of the Magnus and Baker-Cambel-Hausdorff-Dynkin expansions.
result Quantitative estimates between flows and their truncated logarithms.
We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those Riemannian flows M carrying non-trivial solutions to that equation in case M is a local Riemannian product, a Sasakian manifold or 3-dimensional.
The paper estimates curvature for a specific flow on manifolds.
problem Estimating curvature for Ricci-harmonic flow on manifolds.
method Local Lp estimate and De Giorgi-Nash-Moser iteration method. result Local boundedness of Riemannian curvature proved.
Heat flow fails to preserve concavity in curved spaces.
problem Non-preservation of concavity properties in curved spaces.
method Analysis of Dirichlet heat flow on Riemannian manifolds.
result No concavity properties are preserved unless curvature is zero.
The paper proves a Liouville theorem for heat flows on manifolds with specific curvature conditions.
problem Investigating heat flows on manifolds with specific curvature conditions.
method Gradient estimate and Liouville type theorem for ancient solutions.
result Established a Liouville theorem for V-harmonic heat flows. The study applies Riemannian flow theory to Lorentzian manifolds to understand horizons.
problem Understanding the geometry of horizons in Lorentzian manifolds.
method Importing results from Riemannian flows to Lorentzian horizons, clarifying the relation between isometric/geodesible flows and non-degeneracy conditions.
result Theorems on the dynamical structure of compact horizons without relying on degeneracy assumptions.
Proves robust transitivity for geodesic flows from metrics with conjugate points.
problem Transitivity of geodesic flows from metrics with conjugate points.
method General criterion for robust transitivity of partially hyperbolic geodesic flows.
result First example of a C2 open set of Riemannian metrics with conjugate points and transitive geodesic flow. New results on geodesic flows using curve shortening flow.
problem Stability and existence of geodesics in Riemannian surfaces.
method Curve shortening flow approach.
result Existence of infinitely many geodesics intersecting a given one in every primitive class.
We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.
Variational approximations for curve flows on Riemannian manifolds.
problem Approximating solutions to curvature and elastic flow problems on Riemannian manifolds.
method Variational formulations, finite element approximations, piecewise linear elements, stability analysis.
result Derived schemes can compute rotationally symmetric self-shrinkers and geodesics.
Stable discretizations for elastic flow on Riemannian manifolds.
problem Discretizing elastic flow on curved spaces.
method Conformally flat Riemannian manifolds discretization.
result Robust and quadratic convergence of the method.
The paper proves Eells-Sampson type theorems for subelliptic harmonic maps.
problem Existence of subelliptic harmonic maps from sub-Riemannian to Riemannian manifolds.
method Investigates subelliptic harmonic map heat flow under non-positive sectional curvature.
result Proves Eells-Sampson type existence results for subelliptic harmonic maps.
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
The study derives Harnack inequalities for various curvature flows in different spaces.
problem Deriving Harnack inequalities for curvature flows in various spaces.
method Using Harnack estimates and a concept of 'duality' for strictly convex hypersurfaces.
result New Harnack inequalities and pseudo-Harnack inequalities for expanding flows in specific spaces.
Proves stability of geodesic flows on closed surfaces.
problem Stability of geodesic flows on closed surfaces.
method Generic Riemannian metrics and Reeb flows.
result Proves C2-stability conjecture for geodesic flows. Study geodesic flows on cones over Riemannian manifolds, showing superintegrability.
problem Behavior of geodesics on cones over arbitrary Riemannian manifolds.
method Show existence of first integrals uniquely determining geodesics.
result Geodesic flow on cones is superintegrable and Liouville--Arnold integrable for non-radial trajectories.
The study finds complete translating solitons for certain powers of Gaussian curvature in Riemannian products.
problem Exploring translating solitons in Riemannian products with powers of Gaussian curvature.
method Investigating Kα-flows in Riemannian products MimesR for M=Rn,Sn,HFm. result Existence of complete rotational translating solitons for certain values of α in MimesR. Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.
problem Improving convergence rate of Riemannian stochastic gradient descent.
method Using stochastic differential geometry, the paper shows RSGD can be approximated by the Riemannian stochastic modified flow (RSMF).
result RSGD can be approximated by the solution to the RSMF driven by an infinite-dimensional Wiener process, increasing the order of approximation.
Extends heat flow estimates to non-smooth spaces.
problem Heat flow estimates on non-smooth metric measure spaces.
method Extends Hamilton's gradient estimates and monotonicity formula to metric measure spaces.
result Establishes heat flow estimates for metric measure spaces.
Embeds Riemannian manifolds with Anosov flows, linking classical and new theorems.
problem Embedding Riemannian manifolds with Anosov flows.
method Isometric embedding into a closed Riemannian manifold with Anosov geodesic flow.
result Direct link between classical and new theorems.
Study shows how to compute manifold's characteristic numbers from flow properties.
problem Computing characteristic numbers of manifolds.
method Analyzes singular Riemannian flows and their foliations.
result Characteristic numbers computed as residues of infinitesimal foliations.
Exponential rate of convergence for harmonic heat flow maps.
problem Analyzing the convergence rate of harmonic heat flow maps.
method Proving exponential convergence rate for harmonic heat flow maps.
result Exponential convergence rate of the harmonic heat flow.
Study high codimension mean curvature flow in Riemannian manifolds, proving limiting flow in Euclidean space.
problem Analyzing mean curvature flow in high codimension Riemannian manifolds.
method Establishing codimension estimate, using quadratic pinching condition, gradient estimates.
result Existence of limiting flow in Euclidean space under cylindrical pinching condition.
Paper proves Harnack inequalities for Witten Laplacian on manifolds with specific flows.
problem Proving Harnack inequalities for Witten Laplacian on Riemannian manifolds.
method Using Li-Yau and Hamilton type inequalities for heat equation associated with time-dependent Witten Laplacian on manifolds with specific flows.
result Proves Li-Yau and Hamilton type Harnack inequalities for Witten Laplacian.
Geodesic flows with diagonalisable integrals are orthogonal.
problem Understanding geodesic flows with specific integrals.
method Analyzing quadratic integrals for geodesic flows.
result Diagonalisable integrals imply orthogonal separation of variables.
Study inverse problems for twisted geodesic flows on manifolds.
problem Understanding inverse problems for twisted geodesic flows.
method Generalized ray transforms and tensor tomography.
result New insights into rigidity problems for twisted geodesic flows.
The paper studies curves in Riemannian manifolds using total variation flow.
problem Analyzing the evolution of curves in Riemannian manifolds using total variation.
method Defining and proving the existence of strong solutions to the flow equations, showing variational equality, and proving convergence.
result Strong solutions converge to a constant map in finite time for non-positive sectional curvature.
New thermostat flows derived from holomorphic differentials on Riemannian surfaces.
problem Understanding properties of geodesic flows on Riemannian surfaces.
method Introducing a new family of thermostat flows and relating them to weighted holomorphic differentials.
result The introduced flows admit a dominated splitting and can be Anosov under certain conditions.
We give a natural definition of geodesics on a Riemannian supermanifold and extend the usual geodesic flow defined on the cotangent bundle of the body of the supermanifold, associated to the induced Riemannian structure on the body, to a geodesic "superflow" on the cotangent bundle of the supermanifold. Integral curves…
New flows model distributions on Riemannian manifolds without domain knowledge.
problem Limited modeling of distributions on Riemannian manifolds.
method Riemannian convex potential maps using optimal transport.
result These flows can model standard distributions on spheres and tori.
Study sesqui-harmonic map flow from Riemannian surfaces
problem Investigate sesqui-harmonic map flow from Riemannian surfaces
method L2-gradient flow of an energy functional
result Generalizes Struwe's regularity result for harmonic maps
The paper connects geodesic flows, hyperbolic geodesics, and stable ergodicity.
problem Understanding conditions for geodesic flows to be Anosov and ergodic.
method Analyzing Finsler and Riemannian metrics on surfaces, using recent results.
result Geodesic flows on surfaces are C2 stably ergodic if and only if they are Anosov. The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flo…
Study sub-Riemannian structures on Banach manifolds, extending controllability and geodesic flow results.
problem Analyzing sub-Riemannian structures on Banach manifolds.
method Define sub-Riemannian structures, extend Chow-Rashevski theorem, and provide conditions for Hamiltonian geodesic flow.
result Extensions of the Chow-Rashevski theorem for exact controllability and conditions for Hamiltonian geodesic flow in infinite-dimensional setting.
Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.
problem Entropy for submanifolds in Riemannian manifolds.
method Entropy defined and shown to be monotone along mean curvature flow.
result Partial regularity of mean curvature flow limits of surfaces.
We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.
New metrics connect surfaces with Anosov flows to those with negative curvature.
problem Creating metrics with Anosov flows on surfaces of positive curvature.
method Constructing metrics with Anosov geodesic flows and positive curvature regions, connecting to negative curvature metrics via smooth conformal deformations.
result Existence of a smooth curve of conformal deformations connecting Anosov metrics to metrics of negative curvature.
Study of integral flows on Riemannian manifolds with focus on blow-up profiles and concentration-compactness.
problem Analyzing nonlinear integral flows on Riemannian manifolds with specific focus on blow-up profiles and concentration-compactness.
method Investigation of a family of nonlinear integral flows involving Riesz potentials, focusing on the Hardy-Littlewood-Sobolev (HLS) subcritical and critical regimes.
result Established convergence on unit spheres and certain locally conformally flat manifolds for the dual Yamabe flow.
In this paper, we define a class of new geometric flows on a complete Riemannian manifold. The new flow is related to the generalized (third order) Landau-Lifishitz equation. On the other hand it could be thought of a special case of the Schrödinger-Airy flow when the target manifold is a Kähler manifold with constant …