arXiv research

Sharp eigenvalue bounds and splitting for modified Ricci flow.

We study the properties of Modified Riemann extensions evolving under Ricci flow. We obtain the necessary and sufficient condition for modified Riemann extension under Ricci flow to stay as modified Riemann extension. We also discuss the properties of the curvature tensors under Ricci flow.

Modified curve shortening flow constructs $λ$-Angenent curve.

The modified J-flow with Calabi ansatz shows convergence or blow-up behavior based on topological constants.

We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.

We investigate the Kähler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a Kähler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy.

Enhances understanding of Kähler-Ricci flow singularities.

Smooth solutions found for modified mean curvature flow in Riemannian manifolds.

In this note, a modified Kähler-Ricci flow is introduced and studied. The main point is to show the flexibility of Kähler-Ricci flow and summarize some useful techniques.

In this paper the rate relations of Riemann, conformal, conharmonic and Weyl curvature tensors under Yamabe flow are studied. Modified Riemann extensions under Yamabe flow is discussed. The paper ends with remarks on some standard metrics.

We modify the Laplacian coflow of co-closed G2-structures - $\frac{d}{dt}ψ=Δψ$ where $ψ$ is the closed dual 4-form of a $G_{2}$-structure $\varphi$. The modified flow is now parabolic in the direction of closed forms upto diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified f…

We study the convergence of a modified Kaeher-Ricci flow defined by Zhou Zhang. We show that the flow converges to a singular metric when the limit class is degenerate. This proves a conjecture of Zhang.

Smooth convergence to equidistant surfaces in Fuchsian manifolds.

In this paper, we propose a method of studying the modified Kahler-Ricci flow on projective bundles and give the explicit equation from the view point of symplectic geometry.

We show a quite simple second variation formula for Perelman's $\mathcal{W}$-functional along the modified Kähler-Ricci flow over Fano manifolds.

New dynamics for SGD in small learning rate regime.

The paper studies geometric constants under modified Ricci flows with variable parameters.

Let $(\mathcal{M},g)$ be a closed Riemannian manifold. The $\textit{ second order approximation}$ to the perturbative renormalization group flow for the nonlinear sigma model (RG-2 flow) is given by : \[ \frac{\partial }{\partial t} \, g(t) \, =\, -2 \mathrm{Ric}(t) \, -\, \fracα{2} \mathrm{Rm}^2(t), \] where $ g = \ma…

We prove that on a Kähler manifold admitting an extremal metric $ω$ and for any Kähler potential $\varphi_0$ close to $ω$, the Calabi flow starting at $\varphi_0$ exists for all time and the modified Calabi flow starting at $\varphi_0$ will always be close to $ω$. Furthermore, when the initial data is invariant under t…

Study new Einstein-like metrics and their properties.

We survey recent progress in the study of $G_{2}$-structure Laplacian coflows, that is, heat flows of co-closed $G_{2}$-structures. We introduce the properties of the original Laplacian coflow of $G_{2}$-structures as well as the modified coflow, reviewing short-time existence and uniqueness results for the modified co…

The paper studies a modified scalar curvature flow and proves convergence to a sphere.

Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.

This work considers the question of whether mean-curvature flow can be modified to avoid the formation of singularities. We analyze the finite-elements discretization and demonstrate why the original flow can result in numerical instability due to division by zero. We propose a variation on the flow that removes the nu…

Real analyticity proved for modified Laplacian coflow solutions.

Nearly $G_2$-structures are unstable under a modified $G_2$-Laplacian co-flow.

The paper proves the monotonicity of a modified Perelman's W-entropy for mean curvature flow.

Modified Bakry-Émery criterion inequality for Tsallis entropy monotonicity.

We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.

Constructs a moment map flow for isotropic maps on surfaces.

The paper proves stability for a modified Bach flow on various manifolds.

We derive modified Perelman-type monotonicity formulas for solutions to the generalized Ricci flow equation with symmetry on principal bundles, which lead to rigidity and classification results for nonsingular solutions.

We define a new version of modified mean curvature flow (MMCF) in hyperbolic space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural negative $L^2$-gradient flow of the energy functional defined by De Silva and Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the MMCF of com…

New perspective on G2-structures flow from DeTurck Laplacian.

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…

In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to more general flows of G$_2$ 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 G$_2$ structures.

The asymptotic Plateau problem asks for the existence of smooth complete hypersurfaces of constant mean curvature with prescribed asymptotic boundary at infinity in the hyperbolic space $\mathbb{H}^{n+1}$. The modified mean curvature flow (MMCF) was firstly introduced by Xiao and the second author a few years back, and…

In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schrödinger equation, the complex Camassa-Holm equation and the multi-component modified Camassa-Holm equation are pro…

We prove a general result about the stability of geometric flows of "closed" sections of vector bundles on compact manifolds. Our theorem allows to prove a stability result for the modified Laplacian coflow in G2-geometry introduced by Grigorian and for the balanced flow introduced by the authors in a previous paper.

The paper confirms a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.

New method reveals insights about stochastic optimization methods using modified equations.

The study finds infinitely many periodic orbits that can be used to modify Anosov flows.

The paper proves geometric inequalities and their stabilities for curves in hyperbolic space.

The paper studies dimensions of attractors for modified Leray-alpha equation on various surfaces.

Modified dynamical systems retain Turing universality.

We use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.

In this paper, we show that small spherical soap bubbles in irreducible simply connected symmetric spaces of rank greater than one are constructed from the limits of a certain kind of modified mean curvature flows starting from small spheres in the Euclidean space of dimension equal to the rank of the symmetric space, …

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.