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Guoyi Xu

Guoyi Xu appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.DGmath.APComputer Vision

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Published work

8 published item(s)

preprint2026arXiv

The First Controllable Bokeh Rendering Challenge at NTIRE 2026

This study presents the outcomes of the first Controllable Bokeh Rendering Challenge at NTIRE and highlights the most effective submitted methodologies. In total, 44 participants registered for the competition, of which 8 teams submitted valid solutions after the conclusion of the final test phase. All submissions were evaluated on unseen images, focusing on portraits and intricate subjects with complex and visually appealing bokeh phenomena. In addition to the first track focusing on established quantitative fidelity metrics, we conducted a qualitative user study with a panel of experts for a second track focusing on perceptual assessment. As this was the inaugural challenge on this topic, most of the participants focused on refining and extending the Bokehlicious baseline method.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

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.

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preprint2015arXiv

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.

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preprint2013arXiv

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.

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preprint2013arXiv

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.

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preprint2013arXiv

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.

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preprint2013arXiv

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.

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preprint2009arXiv

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 $M^2$.

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