Related papers: Mean Curvature Flow, Orbits, Moment Maps
We prove here that given a proper isometric action $K\times M\to M$ on a complete Riemannian manifold $M$ then every continuous isometric flow on the orbit space $M/K$ is smooth, i.e., it is the projection of an $K$-equivariant smooth flow…
We study the orbit structure and the geometric quantization of a pair of mutually commuting hamiltonian actions on a symplectic manifold. If the pair of actions fulfils a symplectic Howe condition, we show that there is a canonical…
The Skew Mean Curvature Flow(SMCF) is a Schr\"odinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local…
We introduce a variant of Farber's topological complexity, defined for smooth compact orientable Riemannian manifolds, which takes into account only motion planners with the lowest possible "average length" of the output paths. We prove…
We show that whenever a Hamiltonian diffeomorphism or a Reeb flow has a finite number of periodic orbits, the mean indices of these orbits must satisfy a resonance relation, provided that the ambient manifold meets some natural…
Given a smooth Riemannian manifold $(M,g)$, compact and without boundary, we analyze the dynamical optimal mass transport problem where the cost is given by the sum of the kinetic energy and the relative entropy with respect to a reference…
Let $(M,\omega)$ be a symplectic manifold endowed with a agrangian foliation ${\cal L}$, it has been shown by Weinstein [16] hat the symplectic structure of $M$ defines on each leaf of ${\cal L}$, connection which curvature and torsion…
There is an extensive and growing body of work analyzing convex ancient solutions to Mean Curvature Flow (MCF), or equivalently of Rescaled Mean Curvature Flow (RMCF). The goal of this paper is to complement the existing literature, which…
The aim of the paper is three-fold. We begin by proving a formula, both global and local versions, relating the number of periodic orbits of an iterated map and the Lefschetz numbers, or indices in the local case, of its iterations. This…
We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein…
We generalize the notion of calibrated submanifolds to smooth maps and show that the several examples of smooth maps appearing in the differential geometry become the examples of our situation. Moreover, we apply these notion to give the…
Given a compact manifold equipped with a volume element and a Riemannian metric, we formulate and study a dual pair of optimization problems: one concerning smooth maps from the manifold into the Hilbert space $l^2$ and the other concerning…
We consider a Hamiltonian torus action on a compact connected symplectic manifold M. For a certain class of Lagrangian submanifolds Q of M we show that the image of Q under the momentum map is convex. As an application we complete the…
The main point of this paper is that, under suitable conditions on the mean curvature and the Ricci curvature of the ambient space, we can extend Choi-Schoen's Compactness Theorem to compact embedded minimal surfaces to simple immersed…
In this article we give a complete description of the evolution of an area decreasing map $f:M\to N$ induced by its mean curvature in the situation where $M$ and $N$ are complete Riemann surfaces with bounded geometry, $M$ being compact,…
Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group $G$. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers…
We express the mean curvature flow of Lagrangian submanifolds in pseudo-Riemannian manifolds endowed with the Kim-McCann-Warren metric within the framework of generalized mean curvature flow on Kim-McCann manifolds. While generalized mean…
We investigate minimality and stability of periodic brake orbits in natural Lagrangian systems on smooth Riemannian manifolds. We prove that every non-constant periodic brake orbit is not a minimizer of the fixed-time action, for any…
We introduce a class of flows on the Wasserstein space of probability measures with finite first moment on the Cartan-Hadamard Riemannian manifold of positive definite matrices, and consider the problem of differentiability of the…
After reviewing manifold optimization techniques in applications like MIMO communication systems, phased array beamforming, radar, and control theory, we observed that the Complex Circle Manifold (CCM) is widely employed, yet its…