# Hikaru Yamamoto

7.09

· Ph.D. in Mathematical SciencesAbout

11

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Apr 2013 - Mar 2016

Apr 2011 - Mar 2013

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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 establish the theorem, we provide a scale invariant monotone quantity. As a critical point of this quantity, we define self-shrinker solution of the line bundle mean curvature flow. The Liouville type theorem for self-shrinkers is also given. It plays an important role in the proof of the $\varepsilon$-regularity theorem.

- Sep 2018

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 behavior of the solutions near the singular set.

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 prove that, if $1<p<(n-m)/(n-m-2)$, then there are two types of singular solution. Moreover, we show the uniqueness of the solutions and specify the exact behavior of the solutions near the singular set.

- Jul 2017

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 placed for simplicity. In this paper, data to glue their construction on tropical manifolds are proposed and a generalization of the correspondence is proved without the assumption that the Lagrangian submanifold is a section of the torus fibration.

- Feb 2017

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 flow in a Euclidean space by proving its Gauss maps satisfy the harmonic map heat flow equation. In this paper, we deduce the evolution equation for the Gauss maps of a Ricci-mean curvature flow, and as a direct corollary we prove that the Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation when the codimension of submanifolds is 1.

- Aug 2016

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 $\mathbb{P}(\mathcal{O}(0)\oplus \mathcal{O}(k))$ is endowed with the $U(n)$-invariant gradient shrinking Ricci soliton structure. We also prove that there exists a pair of critical radii $r_{1}<r_{2}$ which satisfies the following. The lens space $L(k\, ;1)(r)$ is a self-shrinker if $r<r_{2}$ and self-expander if $r_{2}<r$, and the Ricci-mean curvature flow emanating from $L(k\, ;1)(r)$ collapses to the zero section of $\pi$ if $r<r_{1}$ and to the $\infty$-section of $\pi$ if $r_{1}<r$. This gives explicit examples of Ricci-mean curvature flows.

- May 2015

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 in a Euclidean space.

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 Ricci soliton.

- Nov 2013

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 starting from these examples. We
allow these flows to have some singularities and topological changes.

- Mar 2012

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 of a closed surface of genus g and
the real line R, and a 3 dimensional compact Lagrangian self-shrinker L^2_g in
M_g which is diffeomorphic to the product of the closed surface of genus g and
a circle S^1.

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 singularities of the mean curvature flows. We
also extend the results on special Lagrangian submanifolds on $\mathbb C^n$ to
the toric Calabi-Yau cones over Sasaki-Einstein manifolds.

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