Related papers: Closed warped G$_2$-structures evolving under the …
We consider closed immersed hypersurfaces in $\R^3$ and $\R^4$ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a…
We study Lagrangian immersions in the nearly K\"ahler $\mathbb{S}^6$ which are warped product manifolds of a $1$-dimensional base and a surface. Apart from the totally geodesic ones, they are either of constant sectional curvature…
We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-H\"older-type decomposition of (special)…
In this paper, we study the graphical mean curvature flow in a warped product $_r G/K \times I$, where $G/K$ is a symmetric space of compact type, $I$ is an open interval, and $r$ is a smooth positive function on $I$. If the initial…
This article consists of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. Much of this work has…
We explore three versions of the Laplacian coflow of $G_2$-structures on circle fibrations over Calabi--Yau 3-folds, interpreting their dimensional reductions to the K\"ahler geometry of the base. Precisely, we reduce Ans\"atze for the…
We prove a general criterion to establish existence and uniqueness of a short-time solution to an evolution equation involving "closed" sections of a vector bundle, generalizing a method used recently by Bryant and Xu for studying the…
We extend short-time existence and stability of the Dirichlet energy flow as proven in a previous paper by the authors to a broader class of energy functionals. Furthermore, we derive some monotonely decreasing quantities for the Dirichlet…
We study the gradient flow of Spin($7$)-structures and construct the first explicit solutions, in the homogeneous setting. As an intermediate step, we obtain formulae expressing the Spin($7$)-torsion tensor and gradient flow in terms of the…
We study curve shortening flows in two types of warped product manifolds. These manifolds are $S^1\times N$ with two types of warped metrics where $S^1$ is the unit circle in $R^2$ and $N$ is a closed Riemannian manifold. If the initial…
In this note, we answer affirmatively the question if a warped product of a compact manifold with a line as an ancient solution to the Ricci flow is trivial. We also consider the global behavior of the Type III warping product Ricci flow.
In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows…
Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we…
We introduce geometric flows on a compact almost complex manifold, with the aim to flow a nondegenerate two form to a symplectic two form. We discuss mainly two flows, $d^*d$-flow and $d^*d$-Ricci flow. Among others, we prove the uniqueness…
Among closed G2-structures there are two very distinguished classes: Laplacian solitons and Extremally Ricci-pinched G2-structures. We study the existence problem and explore possible interplays between these concepts in the context of…
The Iwasawa manifold is uplifted to seven-folds of either G_2 holonomy or SU(3) structure, explicit new metrics for the same having been constructed in this work. We uplift the Iwasawa manifold to a G_2 manifold through "size" deformation…
In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the omega-limit of bracket flow solutions…
In this paper, we study the evolution of $L^2$ one forms under Ricci flow with bounded curvature on a non-compact Rimennian manifold. We show on such a manifold that the $L^2$ norm of a smooth one form with compact support is non-increasing…
We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in \cite{ST}. We study the relationship of the existence of the flow and associated static metrics topological information on the…
We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative $L^2$-gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic…