Related papers: A Fourth Order Curvature Flow on a CR 3-manifold
We present several structural results on closed, nonorientable, smooth $4$--manifolds, extending analogous results and machinery for the orientable case. We prove the existence of simplified broken Lefschetz fibrations and simplified…
We prove nonuniqueness results for constant sixth order $Q$-metrics on complete locally conformally flat $n$-dimensional Riemannian manifolds with $n\geqslant 7$. More precisely, assuming a positive Green function exists for the sixth order…
In this paper, we study surfaces which evolve by anisotropic mean curvature flow with contact angle boundary condition over a strictly convex domain in $\mathbb{R}^2$. We establish a prior gradient estimate for smooth solutions to this…
We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\mathbb{R}^n$, intersecting $\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding…
We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $\Sigma$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The…
We study the near-the-interface behavior of a compact convex scalar curvature flow with a flat side. Under suitable initial conditions on the flat side, we show that the interface propagates with a finite and non-degenerate speed until the…
We show the short-time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by volume-preserving mean curvature flow (MCF) taking line tension effects on the boundary into…
$\rm SL(2,\mathbb{C})$ Chern-Simons theory on a closed 3-manifold is one of the most interesting, yet tractable examples of a QFT. On one hand, its non-perturbative structure is not yet fully understood; on the other, the mathematical…
We propose the study of some kind of monopole equations directly associated with a contact structure. Through a rudimentary analysis about the solutions, we show that a closed contact 3-manifold with positive Tanaka-Webster curvature and…
We construct a one-parameter family of solutions to the positive singular Q-curvature problem on compact nondegenerate manifolds of dimension bigger than four with finitely many punctures. If the dimension is at least eight we assume that…
We establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We show that if a contact manifold $(M,\xi)$ admits a hypertight contact form…
This paper concerns the inverse mean curvature flow of convex hypersurfaces which are Lipschitz in general. After defining a weak solution, we study the evolution of the singularity by looking at the blow-up tangent cone around each…
We introduce and study \emph{contact whirl curves} in three-dimensional Lorentzian contact manifolds, with emphasis on the Sasakian setting. This notion refines the concept of whirl curves by encoding the interaction between the adapted…
The main result of this paper is: Given any constant C, there is $(\epsilon,k,L)$ such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a $(\epsilon,k,L)$-neck, then the…
We prove the following result: Let $(X,g_0)$ be a complete, connected 4-manifold with uniformly positive isotropic curvature and with bounded geometry. Then there is a finite collection $\mathcal{F}$ of manifolds of the form $\mathbb{S}^3…
In this article, we prove two "global existence and full convergence theorems" for flow lines of the M\"obius-invariant Willmore flow, and we use these results, in order to prove that fully and smoothly convergent flow lines of the…
The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar three-body problem with both…
Over a bounded strictly convex domain in $\mathbb{R}^n$ with smooth boundary, we establish a priori gradient estimate for an anisotropic mean curvature flow with prescribed contact angle and Neumann boundary conditions. The estimates…
This paper is devoted to studying the initial value problem for a third-order dispersive equation for closed curves into K\"ahler manifolds. This equation is a geometric generalization of a two-sphere valued system modeling the motion of…
Finite topology self translating surfaces to mean curvature flow of surfaces constitute a key element for the analysis of Type II singularities from a compact surface, since they arise in a limit after suitable blow-up scalings around the…