Hikaru Yamamoto

University of Tsukuba · Mathematics

We study the solvability of the initial value problem for the semilinear heat equation ut-Δu=up\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_t-\Delta u=u^p$$\end{do...

We show the noninheritance of the completeness of the noncompact Yamabe flow. Our main theorem states the existence of a long time solution which is complete for each time and converges to an incomplete Riemannian metric. This shows the occurrence of the infinite-time incompleteness.

We can define the “volume” V for Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold X, which can be considered to be the “mirror” of the standard volume for submanifolds. This is called the Dirac-Born-Infeld (DBI) action in physics. In this paper, (1) we introduce the negative gradient flow of V, which we call the l...

We study the solvability of the initial value problem for the semilinear heat equation $u_t-\Delta u=u^p$ in a Riemannian manifold $M$ with a nonnegative Radon measure $\mu$ on $M$ as initial data. We give sharp conditions on the local-in-time solvability of the problem for complete and connected $M$ with positive injectivity radius and bounded sec...

A deformed Donaldson–Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a G2-manifold X satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. The dDT connection is an analogue of a deformed Hermitian Yang–Mills (dHYM) connection which is e...

We show the noninheritance of the completeness of the noncompact Yamabe flow. Our main theorem states the existence of a long time solution which is complete for each time and converges to an incomplete Riemannian metric. This shows the occurrence of the infinite-time incompleteness.

A deformed Donaldson–Thomas connection for a manifold with a \(\mathrm{Spin}(7)\)-structure, which we call a \(\mathrm{Spin}(7)\)-dDT connection, is a Hermitian connection on a Hermitian line bundle L over a manifold with a \(\mathrm{Spin}(7)\)-structure defined by fully nonlinear PDEs. It was first introduced by Lee and Leung as a mirror object of...

We can define the "volume" $V$ for Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold $X$, which can be considered to be the "mirror" of the standard volume for submanifolds. This is called the Dirac-Born-Infeld (DBI) action in physics. In this paper, (1) we introduce the negative gradient flow of $V$, which we call...

The real Fourier-Mukai transform sends a section of a torus fibration to a connection over the total space of the dual torus fibration. By this method, Leung, Yau and Zaslow introduced deformed Hermitian Yang-Mills (dHYM) connections for K\"ahler manifolds and Lee and Leung introduced deformed Donaldson-Thomas (dDT) connections for $G_2$- and ${\rm...

A deformed Donaldson-Thomas connection for a manifold with a ${\rm Spin}(7)$-structure, which we call a ${\rm Spin}(7)$-dDT connection, is a Hermitian connection on a Hermitian line bundle $L$ over a manifold with a ${\rm Spin}(7)$-structure defined by fully nonlinear PDEs. It was first introduced by Lee and Leung as a mirror object of a Cayley cyc...

A deformed Hermitian Yang-Mills (dHYM) connection and a deformed Donaldson-Thomas (dDT) connection are Hermitian connections on a Hermitian vector bundle $L$ over a K\"ahler manifold and a $G_2$-manifold, which are believed to correspond to a special Lagrangian and a (co)associative cycle via mirror symmetry, respectively. In this paper, when $L$ i...

In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given K\"ahler manifold. The goal of this paper is to give an $\varepsilon$-regularity theorem for the line bundle mean curvature flow. To establi...

From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mari\~no, Minasian, Moore and Strominger, and Leung, Yau and Zaslow proved that it naturally appears as a mirror object of special Lagrangian submanifolds via Fourier-Mukai transform between dual torus fibrations. In their paper, some assumptions are...

We consider the heat equation with a superlinear absorption term ∂tu−Δu=−up in Rⁿ and study the existence of nonnegative solutions with an m-dimensional time-dependent singular set, where n−m≥3. We prove that if 1<p<(n−m)/(n−m−2), then there are two types of singular solutions. Moreover, we show the uniqueness of the solutions and specify the exact...

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended this result to a mean curvature f...

We consider the heat equation with a superlinear absorption term $\partial_{t} u-\Delta u= -u^{p}$ in $\mathbb{R}^n$ and study the existence and nonexistence of nonnegative solutions with an $m$-dimensional time-dependent singular set, where $n-m\geq 3$. First, we prove that if $p\geq (n-m)/(n-m-2)$, then there is no singular solution. We next prov...

In this paper, we give a lower bound estimate for the diameter of a Lagrangian self-shrinker in a gradient shrinking K\"ahler-Ricci soliton as an analog of a result of A. Futaki, H. Li and X.-D. Li for a self-shrinker in a Euclidean space. We also prove an analog of a result of H.-D. Cao and H. Li about the non-existence of compact self-expanders i...

Let $\pi:\mathbb{P}(\mathcal{O}(0)\oplus \mathcal{O}(k))\to \mathbb{P}^{n-1}$ be a projective bundle over $\mathbb{P}^{n-1}$ with $1\leq k \leq n-1$. In this paper, we show that lens space $L(k\, ;1)(r)$ with radius $r$ embedded in $\mathbb{P}(\mathcal{O}(0)\oplus \mathcal{O}(k))$ is a self-similar solution, where $\mathbb{P}(\mathcal{O}(0)\oplus \...

Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize this result for a Ricci-mean curvature flow moving along a Ricci flow constructed from a gradient shrinking Ri...

In this paper we generalize examples of Hamiltonian stationary Lagrangian submanifolds constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau manifolds. We construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized Lagrangian mean curvature flows...

We construct some examples of special Lagrangian submanifolds and Lagrangian self-similar solutions in almost Calabi-Yau cones over toric Sasaki manifolds. For example, for any integer g>0, we can construct a real 6 dimensional Calabi-Yau cone M_g and a 3 dimensional special Lagrangian submanifold L^1_g in M_g which is diffeomorphic to the product...

The self-similar solutions to the mean curvature flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asymptotic behavior for the singul...