arXiv subjects

Guoyi Xu

Publications and source records attributed to Guoyi Xu.

45 records · Page 3Linked to original sources

arXiv2016

The (p;m)-width of Riemannian manifolds and its realization

While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Marques and Neves used the $p$-width to prove the existence of infinite many minimal hypersurfaces in compact manifolds with positive Ricci curvature. However, whether the $p$-width can be realized as the volume of minimal hypersurfaces is not known yet. We introduced the concept of the $(p, m)$-width which can be viewed as the stratification of the $p$-width, and proved that the $(p, m)$-width can be realized as the volume of minimal hypersurfaces with multiplicities.

math.DG↗

arXiv2016

Three circles theorems for harmonic functions

We proved two Three Circles Theorems for harmonic functions on manifolds in integral sense. As one application, on manifold with nonnegative Ricci curvature, whose tangent cone at infinity is the unique metric cone with unique conic measure, we showed the existence of nonconstant harmonic functions with polynomial growth. This existence result recovered and generalized the former result of Y. Ding, and led to a complete answer of L. Ni's conjecture. Furthermore in similar context, combining the techniques of estimating the frequency of harmonic functions with polynomial growth, which were developed by Colding and Minicozzi, we confirmed their conjecture about the uniform bound of frequency.

math.DG↗

arXiv2015

Lower bound of Ricci flow's existence time

Let $(M^n, g)$ be a compact $n$-dim ($n\geq 2$) manifold with nonnegative Ricci curvature, and if $n\geq 3$ we assume that $(M^n, g)\times \mathbb{R}$ has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on $(M^n, g)$ is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows' maximal existence times, which was firstly proved by E. Cabezas-Rivas and B. Wilking. We also get an interior curvature estimates for $n= 3$ under $Rc\geq 0$ assumption among others. Combining these results, we proved the short time existence of the Ricci flow on a large class of $3$-dim open manifolds, which admit some suitable exhaustion covering and have nonnegative Ricci curvature.

math.DG↗

arXiv2014

An equation linking $\mathscr{W}$-entropy with reduced volume

$\mathscr{W}$-entropy and reduced volume for the Ricci flow were introduced by Perelman, which had proved their importance in the study of the Ricci flow. L. Ni studied the analogous concepts for the linear heat equation on the static manifolds, and established an equation which links the large time behavior of these two. Due to the surprising similarity between those concepts in the Ricci flow and the linear heat equation, a natural question whether such equation holds for the Ricci flow ancient solution was asked by L. Ni. In this paper, we gave an alternative proof to L. Ni's original equation based on a new method. And following the same philosophy of this method, we answer L. Ni's question positively for Type I $κ$-solutions of the Ricci flow.

math.DG↗

arXiv2013

Large time behavior of the heat kernel

In this paper, we study the large time behavior of the heat kernel on complete Riemannian manifolds with nonnegative Ricci curvature, which was studied by P. Li with additional maximum volume growth assumption. Following Y. Ding's original strategy, by blowing down the metric, using Cheeger and Colding's theory about limit spaces of Gromov-Hausdorff convergence, combining with the Gaussian upper bound of heat kernel on limit spaces, we succeed in reducing the limit behavior of the heat kernel on manifold to the values of heat kernels on tangent cones at infinity of manifold with renormalized measure. As one application, we get the consistent large time limit of heat kernel in more general context, which generalizes the former result of P. Li. Furthermore, by choosing different sequences to blow down the suitable metric, we show the first example manifold whose heat kernel has inconsistent limit behavior, which answers an open question posed by P. Li negatively.

math.DG↗

arXiv2013

Local pinching estimates in 3-dim Ricci flow

We study curvature pinching estimates of Ricci flow on complete 3- dimensional manifolds without bounded curvature assumption. We will derive some general curvature conditions which are preserved on any complete solution of 3-dim Ricci flow, these conditions include nonnegative Ricci curvature and sectional curvature as special cases. A local version of Hamilton-Ivey estimates is also obtained.

math.DG↗

arXiv2013

The short time asymptotics of Nash entropy

Let $(M^n, g)$ be a complete Riemannian manifold with $Rc\geq -Kg$, $H(x, y, t)$ is the heat kernel on $M^n$, and $H= (4πt)^{-\frac{n}{2}}e^{-f}$. Nash entropy is defined as $N(H, t)= \int_{M^n} (fH) dμ(x)- \frac{n}{2}$. We studied the asymptotic behavior of $N(H, t)$ and $\frac{\partial}{\partial t}\Big[N(H, t)\Big]$ as $t\rightarrow 0^{+}$, and got the asymptotic formulas at $t= 0$. In the Appendix, we got Hamilton-type upper bound for Laplacian of positive solution of the heat equation on such manifolds, which has its own independent interest.

math.DG↗

arXiv2013

Short-time existence of the Ricci flow on noncompact Riemannian manifolds

In this paper, we give the first detailed proof of the short-time existence of Deane Yang's local Ricci flow. Then using the local Ricci flow, we prove short-time existence of the Ricci flow on noncompact manifolds, whose Ricci curvature has global lower bound and sectional curvature has only local average integral bound. The short-time existence of the Ricci flow on noncompact manifolds with bounded curvature was studied by Wan-Xiong Shi in 1990s. As a corollary of our main theorem, we get the short-time existence part of Shi's theorem in this more general context.

math.DG↗

arXiv2009

Examples of hypersurfaces flowing by curvature in a Riemannian manifold

This paper gives some examples of hypersurfaces $ϕ_t(M^n)$ evolving in time with speed determined by functions of the normal curvatures in an $(n+1)$-dimensional hyperbolic manifold; we emphasize the case of flow by harmonic mean curvature. The examples converge to a totally geodesic submanifold of any dimension from 1 to $n$, and include cases which exist for infinite time. Convergence to a point was studied by Andrews, and only occurs in finite time. For dimension $n=2,$ the destiny of any harmonic mean curvature flow is strongly influenced by the genus of the surface $M^2$.

math.DG↗