Related papers: Gradient Flows for Dirichlet Eigenvalues

We prove an abstract compactness result for gradient flow lines of a non-local unregularized gradient flow equation on a scale Hilbert space. This is the first step towards Floer theory on scale Hilbert spaces.

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of…

We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the…

We explicitly construct parameter transformations between gradient flows in metric spaces, called curves of maximal slope, having different exponents when the associated function satisfies a suitable convexity condition. These…

We consider the (sub-Riemannian type) control problem of finding a path going from an initial point $x$ to a target point $y$, by only moving in certain admissible directions. We assume that the corresponding vector fields satisfy the…

Gradient descent-ascent (GDA) flows play a central role in finding saddle points of bivariate functionals, with applications in optimization, game theory, and robust control. While they are well-understood in Hilbert and Banach spaces via…

Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special…

We establish convergence results for a spatial semidiscretization of Mean Curvature Flow (MCF) for surfaces with fixed boundaries. Our analysis is based on Huisken's evolution equations for the mean curvature and the normal vector, enabling…

In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, $L^p$-inequalities and…

We consider a gradient flow associated to the mean field equation on $(M,g)$ a compact riemanniann surface without boundary. We prove that this flow exists for all time. Moreover, letting $G$ be a group of isometry acting on $(M,g)$, we…

We study the $L^2$-gradient flow of functionals $\mathcal F$ depending on the eigenvalues of Schr\"odinger potentials $V$ for a wide class of differential operators associated to closed, symmetric, and coercive bilinear forms, including the…

The mean curvature flow describes the parabolic deformation of embedded branes in Riemannian geometry driven by their extrinsic mean curvature vector, which is typically associated to surface tension forces. It is the gradient flow of the…

In [8], the gradient conjecture of R. Thom was proven for gradient flows of analytic functions on Rn. This result means that the secant at a limit point converges, so that the flow cannot spiral forever. Once the trajectory becomes…

We develop the theory of discrete-time gradient flows for convex functions on Alexandrov spaces with arbitrary upper or lower curvature bounds. We employ different resolvent maps in the upper and lower curvature bound cases to construct…

We consider the gradient flow structure of the porous medium equations with non-negative constant Dirichlet boundary conditions. We construct weak solutions to the equations via the minimizing movement scheme by considering an entropy…

For large classes of non-convex subsets $Y$ in ${\mathbb R}^n$ or in Riemannian manifolds $(M,g)$ or in RCD-spaces $(X,d,m)$ we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space $(Y,d_Y,m_Y)$…

We prove lower bounds for the Dirichlet Laplacian on possibly unbounded domains in terms of natural geometric conditions. This is used to derive uncertainty principles for low energy functions of general elliptic second order divergence…

Consider an arbitrary closed, countably $n$-rectifiable set in a strictly convex $(n+1)$-dimensional domain, and suppose that the set has finite $n$-dimensional Hausdorff measure and the complement is not connected. Starting from this given…

In this paper, we introduce a new method to establish existence of geometric flows with surgery. In contrast to all prior constructions of flows with surgery in the literature our new approach does not require any a priori estimates in the…

In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential…