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Xuan Yao

Publications and source records attributed to Xuan Yao.

30 records · Page 2Linked to original sources
arXiv2025

Euclidean Domains with Nearly Maximal Yamabe Quotient

Let $Ω$ be a smooth, bounded domain in $\mathbb R^3$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $Ω$ is at most the Yamabe quotient of a ball, and equality holds if and only if $Ω$ is a ball. We show that if equality almost holds then the following things are true: (i)$Ω$ is diffeomorphic to a ball; (ii) There is a small number $ε> 0$ such that $B(x,r) \subset Ω\subset B(x,r(1+ε))$; (iii) After suitable scaling, $Ω$ is Gromov-Hausdorff close to the unit ball when considered as a metric space with its induced length metric. We also give a qualitative comparison between $Q$ and the coefficient of quasi-conformality studied in the theory of quasi-conformal maps.

math.DG↗
arXiv2024

A Note on Scalar curvature comparison rigidity for compact domains

We prove a generalization of Gromov's conjecture on scalar curvature rigidity of convex polytopes to arbitrary convex Riemannian polytope type domains via harmonic spinors on convex domians with boundary condition constructed by Brendle. In particular, we prove a rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain in Euclidean space, which is a parallel of Shi-Tam's results.

math.DG↗
arXiv2024

Mass, Conformal Capacity, and the Volumetric Penrose Inequality

Let $Ω$ be a smooth, bounded subset of $\mathbb{R}^3$ diffeomorphic to a ball. Consider $M = \mathbb{R}^3 \setminus Ω$ equipped with an asymptotically flat metric $g = f^4 g_{\text{euc}}$, where $f\to 1$ at infinity. Assume that $g$ has non-negative scalar curvature and that $Σ= \partial M$ is a minimal 2-sphere in the $g$ metric. We prove a sharp inequality relating the ADM mass of $M$ with the conformal capacity of $Ω$. As a corollary, we deduce a sharp lower bound for the ADM mass of $M$ in terms of the Euclidean volume of $Ω$. We also prove a stability type result for this ``volumetric Penrose inequality.'' The proofs are based on a monotonicity formula holding along the level sets of a 3-harmonic function.

math.DG↗
arXiv2024

Minimal surfaces with low genus in lens spaces

Given a Riemannian $\mathbb{RP}^3$ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or exist one minimal real projective plane together with two distinct minimal $2$-spheres. Our proof is based on a variant multiplicity one theorem for the Simon-Smith min-max theory under certain equivariant settings. In particular, we show under the positive Ricci assumption that $\mathbb{RP}^3$ contains at least four distinct minimal real projective planes and four distinct minimal tori. Additionally, the number of minimal tori can be improved to five for a generic positive Ricci metric on $\mathbb{RP}^3$ by the degree method. Moreover, using the same strategy, we show that in the lens space $L(4m,2m\pm 1)$, $m\geq 1$, with a bumpy metric or a metric of positive Ricci curvature, there either exist $N(m)$ numbers of distinct minimal Klein bottles, or exist one minimal Klein bottle and three distinct minimal $2$-spheres, where $N(1)=4$, $N(m)=2$ for $m\geq 2$, and the first case happens under the positive Ricci assumption.

math.DG↗
arXiv2024

Fast-Slow Test-Time Adaptation for Online Vision-and-Language Navigation

The ability to accurately comprehend natural language instructions and navigate to the target location is essential for an embodied agent. Such agents are typically required to execute user instructions in an online manner, leading us to explore the use of unlabeled test samples for effective online model adaptation. However, for online Vision-and-Language Navigation (VLN), due to the intrinsic nature of inter-sample online instruction execution and intra-sample multi-step action decision, frequent updates can result in drastic changes in model parameters, while occasional updates can make the model ill-equipped to handle dynamically changing environments. Therefore, we propose a Fast-Slow Test-Time Adaptation (FSTTA) approach for online VLN by performing joint decomposition-accumulation analysis for both gradients and parameters in a unified framework. Extensive experiments show that our method obtains impressive performance gains on four popular benchmarks. Code is available at https://github.com/Feliciaxyao/ICML2024-FSTTA.

cs.CV↗
arXiv2024

On the stability of the Yamabe invariant of $S^3$

Let $g$ be a complete, asymptotically flat metric on $\mathbb{R}^3$ with vanishing scalar curvature. Moreover, assume that $(\mathbb{R}^3,g)$ supports a nearly Euclidean $L^2$ Sobolev inequality. We prove that $(\mathbb{R}^3,g)$ must be close to Euclidean space with respect to the $d_p$-distance defined by Lee-Naber-Neumayer. We then discuss some consequences for the stability of the Yamabe invariant of $S^3$. More precisely, we show that if such a manifold $(\mathbb{R}^3,g)$ carries a suitably normalized, positive solution to $Δ_g w + λw^5 = 0$ then $w$ must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere.

math.DG↗
arXiv2024

Monotone Quantities for $p$-Harmonic functions and the Sharp $p$-Penrose inequality

Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $Σ$. Fix a number $1 < p < 3$. We derive monotone quantities for $p$-harmonic functions on $M$ which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of $M$ with the $p$-capacity of $Σ$ in $M$, which was first proved by Xiao using weak inverse mean curvature flow.

math.DG↗
arXiv2023

Generalized $S^1$-stability theorem

We use the equivariant $μ$-bubbles technique to prove that for any compact manifold $M^n$ with non-empty boundary, $n\in\{3,5,6\}$, the Yamabe invariant of $M^n$ is positive if and only if the Yamabe invariant of $M^n\times S^1$ is positive. This generalized the $S^1$-stability conjecture of Rosenberg to compact manifolds with boundary.

math.DG↗
arXiv2023

On the Yamabe invariant of certain compact manifolds with boundary

We generalize Kobayashi's connected-sum inequality to the $λ$-Yamabe invariants. As an application, we calculate the $λ$-Yamabe invariants of $\#m_1\mathbb{RP}^n\# m_2(\mathbb{RP}^{n-1}\times S^1)\#lH^n\#kS_+^n$, for any $λ\in [0,1]$, $n\geq 3$, provided $k+l\geq 1$. As a corollary, we prove that $\mathbb{RP}^n$ minus finitely many disjoint $n$-balls have the same $λ$-Yamabe invariants as the hemi-sphere, which forms an interesting contrast with the famous Bray-Neves results on the Yamabe invariants of $\mathbb{RP}^3$.

math.DG↗
arXiv2022

Medical Matting: A New Perspective on Medical Segmentation with Uncertainty

It is difficult to accurately label ambiguous and complex shaped targets manually by binary masks. The weakness of binary mask under-expression is highlighted in medical image segmentation, where blurring is prevalent. In the case of multiple annotations, reaching a consensus for clinicians by binary masks is more challenging. Moreover, these uncertain areas are related to the lesions' structure and may contain anatomical information beneficial to diagnosis. However, current studies on uncertainty mainly focus on the uncertainty in model training and data labels. None of them investigate the influence of the ambiguous nature of the lesion itself.Inspired by image matting, this paper introduces alpha matte as a soft mask to represent uncertain areas in medical scenes and accordingly puts forward a new uncertainty quantification method to fill the gap of uncertainty research for lesion structure. In this work, we introduce a new architecture to generate binary masks and alpha mattes in a multitasking framework, which outperforms all state-of-the-art matting algorithms compared. The proposed uncertainty map is able to highlight the ambiguous regions and a novel multitasking loss weighting strategy we presented can improve performance further and demonstrate their concrete benefits. To fully-evaluate the effectiveness of our proposed method, we first labelled three medical datasets with alpha matte to address the shortage of available matting datasets in medical scenes and prove the alpha matte to be a more efficient labeling method than a binary mask from both qualitative and quantitative aspects.

cs.CV↗