The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.
problem Understanding the behavior of parabolic frequency under Ricci flow and Ricci-harmonic flow.
method Investigates the monotonicity of parabolic frequency for solutions of linear and heat equations with bounded curvatures.
result Establishes monotonicity results for parabolic frequency under specific curvature conditions.
Study on biharmonic map heat flow with monotonicity formula.
problem Properties of biharmonic heat kernel and extrinsic biharmonic map heat flow.
method Derived an entropy type quantity exhibiting monotonicity behaviors.
result Monotonicity formula for extrinsic biharmonic map heat flow.
A local monotonicity formula for the Yang-Mills-Higgs flow on G-bundles over Rn (n>4) is proved. It is shown that the monotone quantity coïncides on certain self-similar solutions with that appearing in existing non-local monotonicity formulæ for the Yang-Mills and Yang-Mills-Higgs flows.
Paper defines parabolic frequency for Ricci flow solutions, proving monotonicity and uniqueness.
problem Backwards uniqueness for solutions of parabolic equations on Ricci flows.
method Defines and proves monotonicity of parabolic frequency for Ricci flow solutions.
result Backwards uniqueness for solutions of parabolic equations on Ricci flows.
Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
In this paper, we study the relation of the monotonicity of Hawking Mass and geometric flow problems. We show that along the Hamilton-DeTurck flow with bounded curvature coupled with the modified mean curvature flow, the Hawking mass of the hypersphere with a sufficiently large radius in Schwarzschild spaces is monoton…
The paper proves the monotonicity of a modified Perelman's W-entropy for mean curvature flow.
problem Proving the monotonicity of Perelman's W-entropy for mean curvature flow.
method Modified definition of K. Ecker's W-entropy and Hamilton's Harnack inequality.
result The modified W-entropy is monotonically decreasing in time.
New formulas derived for scalar curvature in generalized Ricci flow.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.
problem Frequency monotonicity for positive solutions of nonlinear equations under Ricci flow.
method Obtained parabolic frequency monotonicity for solutions of two nonlinear parabolic equations with bounded Ricci curvature.
result Established integral type Harnack inequalities using parabolic frequency monotonicity.
In this paper, we introduce a monotonicity formula for the mean curvature flow. We also apply this monotonicity formula to study the asymptotic behavior of eternal solutions.
The paper defines a frequency for mean curvature flow and proves its monotonicity.
problem Backwards uniqueness for solutions of mean curvature flow.
method Defining and proving monotonicity of a parabolic frequency for mean curvature flows.
result Frequency monotonicity implies backwards uniqueness for mean curvature flow solutions.
We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken's monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is…
We construct a class of monotonic quantities along the normalized Ricci flow on closed n-dimensional manifolds.
The paper studies geometric constants under modified Ricci flows with variable parameters.
problem Understanding geometric constants under variable coupling parameters in Ricci flows.
method Introduced modified Ricci flows with variable coefficients, derived evolution formulas, and proved monotonicity conditions.
result Conditions for maintaining monotonicity of geometric constants under modified Ricci flows.
In this paper we introduce the log entropy functional and establish its monotonicity along the Ricci flow. One consequence of it is the monotonicity of the logarithmic Sobolev constant along the Ricci flow.
In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem \ref{theo rescaled}.
We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in Cn.
New proof of energy functional monotonicity via geodesics in measure space.
problem Proving monotonicity of energy functional in generalized Ricci flow.
method Defining adapted cost functional, geodesics, and entropy functional.
result Monotonicity of cost along backwards heat flow and energy functional along generalized Ricci flow.
The paper extends the avoidance principle for mean curvature flows, proving new intersection dimension monotonicity results.
problem Understanding the behavior of intersections in mean curvature flows.
method Proving new intersection dimension monotonicity results for mean curvature flows, Brakke flows, and level set flows.
result The dimension of the intersection of mean curvature flows is non-increasing over time.
Study shows rigidity for entropy minimizers in non-monotone cases.
problem Rigidity of entropy minimizers in non-monotone settings.
method Elementary proofs in non-monotone situations.
result Showed rigidity for minimizers of generalized Colding-Minicozzi entropies.
In this paper, we introduce a monotonicity formula for the mean curvature flow which is related to self-expanders. Then we use the monotonicity to study the asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces.
Unified view of monotonicity formulas for inverse mean curvature flow and p-capacitary potentials.
problem Understanding monotonicity formulas for various geometric flows and potentials.
method Refined analysis of p-capacitary potentials and their level sets. result Strong convergence of p-capacitary potentials to inverse mean curvature flow and curvature varifolds. We derive identities for general flows of Riemannian metrics that may be regarded as local mean-value, monotonicity, or Lyapunov formulae. These generalize previous work of the first author for mean curvature flow and other nonlinear diffusions. Our results apply in particular to Ricci flow, where they yield a local mo…
Proves estimates for Kähler-Ricci flow solutions.
problem Positive solutions to Kähler-Ricci flow.
method Matrix Li-Yau-Hamilton estimates coupled with flow.
result Monotonicity formula derived.
The paper studies gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.
problem Gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.
method Gradient estimates and Harnack inequalities for heat equations under the Laplacian G_2 flow.
result Monotonicity of parabolic frequency and backward uniqueness for positive solutions.
New quantity helps map homotopy classes in complex spaces.
problem Understanding homotopic classes of maps between complex spaces.
method Identified a new monotone quantity in mean curvature flows of maps between Riemannian manifolds.
result Sharp criteria for homotopic classes of maps between complex projective spaces and spheres.
In this paper, we derive the evolution equation for the first eigenvalue of the Witten-Laplace operator acting on the space of functions along the mean curvature flow on a closed oriented manifold. We show some interesting monotonic quantities under the mean curvature flow.
Paper proves estimates for heat and conjugate heat equations under Ricci flow, leading to monotonicity of parabolic frequencies.
problem Establishing estimates for heat and conjugate heat equations under Ricci flow.
method Proving matrix Li-Yau-Hamilton estimates for positive solutions to the heat and conjugate heat equations coupled with Ricci flow.
result Monotonicity of parabolic frequencies established up to correction factors.
New PDE systems generalize Hawking mass monotonicity.
problem Generalizing Hawking mass monotonicity to initial data sets.
method Introduced new systems of PDE on initial data sets (M,g,k). result Generalized Geroch's monotonicity formula to initial data sets.
Curve shortening flow increases annulus modulus.
problem Behavior of annulus modulus under curve shortening flow.
method Nested curves evolving under curve shortening flow.
result Modulus of enclosed annulus is monotonically increasing.
Study nondegenerate singularities in mean curvature flow.
problem Understanding the behavior of nondegenerate cylindrical singularities.
method New L2-distance monotonicity formula and discrete almost monotonicity. result Topology change agrees with level sets change near a critical point of a Morse function.
Proves volume comparison and monotonicity for Bakry-Émery Ricci curvature.
problem Volume comparison and monotonicity for Bakry-Émery Ricci curvature.
method Relative volume comparison theorem for LP-bound of Bakry-Émery Ricci curvature and gradient of potential function. result Modified proof for volume comparison and monotonicity of Kähler-Ricci flow.
We give a family of monotone quantities along smooth solutions to the inverse curvature flows in Euclidean spaces. We also derive a related geometric inequality for closed hypersurfaces with positive k-th mean curvature.
We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter be…
In this paper, we study monotonicity formulas of eigenvalues and entropies along the rescaled List's extended Ricci flow. We derive some monotonicity formulas of eigenvalues of Laplacian which generalize those of Li in [8] and Cao-Hou-Ling in [3]. Moreover, we also consider monotonicity formulas of Fk-func…
We show a connection between the linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow and the monotonicity formula for the positive currents. As an application of the linear trace Li-Yau-Hamilton stated in this paper and the one proved by Chow-Hamiltonm, we give another proof on the classification of the …
In this paper, we extend the Hamilton's gradient estimates \cite{har93} and a monotonicity formula of entropy \cite{ni04} for heat flows from smooth Riemannian manifolds to (non-smooth) metric measure spaces with appropriate Riemannian curvature-dimension condition.
In this article, we introduce a new type of mean curvature flow for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume is a constant and the surface area evolves monotonically. Moreover, for a bounde…
Proves flows of two-convex Lagrangians are regular, global, and converge.
problem Proves regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case.
method Uses a newly discovered monotone quantity to control two-convexity.
result Proves results for the mean curvature flow of area-decreasing Lagrangian submanifolds.
Study applies Huisken formula to mean curvature flow in Ricci soliton background.
problem Analyzing mean curvature flow in Ricci soliton backgrounds.
method Applies Huisken's monotonicity formula to a shrinking self-similar solution of the extended Ricci flow.
result Establishes new results and solves noncompact case under natural geometric assumptions.
Stabilization technique applied to curve shortening flow in 3D space.
problem Stabilizing curve shortening flow in 3D space.
method Applying stabilization technique developed by T. Zelenyak to curve shortening flow in R3. result Derivation of several new monotonicity formulas for curve shortening flow.
We prove that monotonicity of density and energy inequality imply the rectifiability of the singular sets for Yang-Mills flow.
We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in Rn. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypers…
Develops methods to analyze feature-outcome associations in subpopulations.
problem Challenges in understanding feature-outcome associations in high-dimensional data.
method Geometric decomposition framework using gradient flow and co-monotonicity decomposition.
result Identifies context-dependent patterns and improves statistical power and interpretability.
Develops a dynamical method to prove the sharp Berezin-Li-Yau inequality.
problem Proving the sharp Berezin-Li-Yau inequality for convex domains.
method Volume-preserving mean curvature flow and a new monotonicity principle.
result Shows the sharp Berezin-Li-Yau bound for every smooth convex domain.
The Cheeger constant increases under Ricci flow on spheres.
problem Behavior of the Cheeger constant under Ricci flow.
method Evolution identities for parallel curves and viscosity formulation of logh. result The Cheeger constant is non-decreasing under Ricci flow on surfaces diffeomorphic to S2. In this paper, we study monotonicity of eigenvalues of Laplacian-type operator −Δ+cR, where c is a constant, along the Ricci-Bourguignon flow. For c=0, We derive monotonicity of the lowest eigenvalue of Laplacian-type operator −Δ+cR which generalizes some results of Cao \cite{Cao2007}. For c=0, We derive m…
Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. R…