arXiv subjects
Publications and source records attributed to Keita Kunikawa.
We construct an embedded torus and an entire graph in $\mathbb{R}^4$ whose normal bundles are initially flat but lose this property instantaneously under mean curvature flow. We also give an example showing that the parallel principal normal condition is not preserved under mean curvature flow, even though the normal bundle remains flat along the flow.
We show that the Morse index of unstable closed minimal hypersurface $Σ$ in a compact semi-simple Riemannian symmetric space $M=G/K$ is bounded from below by constant times the first Betti number of $Σ$. Our proof is based on a natural extension of the previous method and this also provides a novel approach for the index estimate.
The aim of this paper is to study almost rigidity properties of super Ricci flow whose Muller quantity is non-negative. We conclude almost splitting and quantitative stratification theorems that have been established by Bamler for Ricci flow. As a byproduct, we obtain an almost constancy for a certain integral quantity concerning scalar curvature at an almost selfsimilar point, which is new even for Ricci flow.
Bamler-Zhang have developed geometric analysis on Ricci flow with scalar curvature bound. The aim of this paper is to extend their work to various geometric flows. We generalize some of their results to super Ricci flow whose Muller quantity is non-negative, and obtain Gaussian heat kernel estimates.
We study harmonic map heat flow along ancient super Ricci flow, and derive several Liouville theorems with controlled growth from Perelman's reduced geometric viewpoint. For non-positively curved target spaces, our growth condition is sharp. For positively curved target spaces, our Liouville theorem is new even in the static case (i.e., for harmonic maps); moreover, we point out that the growth condition can be improved, and almost sharp in the static case. This fills the gap between the Liouville theorem of Choi and the example constructed by Schoen-Uhlenbeck.
In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation.
In this article, we establish a monotonicity formula of Hamilton type entropy along Ricci flow on compact surfaces with boundary. We also study the relation between our entropy functional and the $\mathcal{W}$-functional of Perelman type.
The aim of this article is to provide a Liouville theorem for heat equation along ancient super Ricci flow. We formulate such a Liouville theorem under a growth condition concerning Perelman's reduced distance.
We point out that Ecker's local integral quantity agrees with Huisken's global integral quantity at infinity for ancient mean curvature flows if Huisken's one is finite on each time-slice. In particular, this means that the finiteness of Ecker's integral quantity at infinity implies the finiteness of the entropy at infinity.
Inspired by the work of Leung-Wan, we study the mean curvature flow in hyperkähler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold, we define a new energy concept called the "twistor energy" by means of the associated twistor family (i.e. 2-sphere of complex structures). We will show that the mean curvature flow starting at any hyper-Lagrangian submanifold with sufficiently small twistor energy will exist for all time and converge to a complex Lagrangian submanifold for one of the hyperkähler complex structure. In particular, our result implies some kind of energy gap theorem for hyperkähler manifolds which have no complex Lagrangian submanifolds.
In this paper, we derive a mean curvature estimate for eternal solutions (including translating solutions) of almost-calibrated Lagrangian mean curvature flow in complex Euclidean space. As a consequence, we show a non-existence result for eternal solutions of almost-calibrated Lagrangian mean curvature flow.
We show that any complete $f$-stable translating soliton $M$ admits no codimension one cycle which does not disconnect $M$. As a corollary, it follows that any two dimensional complete $f$-stable translating soliton has genus zero.
In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a Kähler-Einstein manifold to more general Kähler manifolds including a Fano manifold equipped with a Kähler form $ω\in 2πc_1(M)$ by using the methodology proposed by T. Behrndt. Namely, we first consider a weighted measure on a Lagrangian submanifold $L$ in a Kähler manifold $M$ and investigate the variational problem of $L$ for the weighted volume functional. We call a stationary point of the weighted volume functional $f$-minimal, and define the notion of Hamiltonian $f$-stability as a local minimizer under Hamiltonian deformations. We show such examples naturally appear in a toric Fano manifold. Moreover, we consider the generalized Lagrangian mean curvature flow in a Fano manifold which is introduced by Behrndt and Smoczyk-Wang. We generalize the result of H. Li, and show that if the initial Lagrangian submanifold is a small Hamiltonian deformation of an $f$-minimal and Hamiltonian $f$-stable Lagrangian submanifold, then the generalized MCF converges exponentially fast to an $f$-minimal Lagrangian submanifold.