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Yuxin Ge

Publications and source records attributed to Yuxin Ge.

At least 19 recordsLinked to original sources
arXiv2026

Optimal geometric inequalities and fully nonlinear conformal flows

We establish sharp Sobolev-type geometric inequalities on $\mathbb{S}^n$ involving the total $\sigma_k$-curvatures $\int_{\mathbb{S}^n}\sigma_k(g)\,dv_g$. These results extend the optimal inequalities of Guan--Wang~\cite{GWDuke} from the cone $\mathcal{C}_k$ to the strictly larger cone $\mathcal{C}_{k-1}$, thereby enlarging the range of admissible conformal metrics. Our approach is variational and is implemented through a fully nonlinear conformal flow. Working in $\mathcal{C}_{k-1}$ introduces substantial analytic difficulties; in particular, one must obtain $C^2$ a priori estimates while simultaneously verifying that the flow remains parabolic. We resolve these issues via a carefully designed test function and by applying the maximum principle to the maximal eigenvalue of the Hessian matrix. As applications, we solve two open problems in dimensions 3 and 4. Finally, we give examples to show that these inequalities cannot be extended to $\mathcal{C}_{k-2}$.

math.DG↗
arXiv2026

The $\sigma_k$-Yamabe problem revisited

In this paper we revisit the $\sigma_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $\sigma_k$-scalar curvature. We prove that on a closed manifold $\left(M,\left[g_0\right]\right)$ with positive Yamabe constant $Y_1\left(M,\left[g_0\right]\right)>0$, the $\sigma_2$-Yamabe constant $$ Y_2\left(M,\left[g_0\right]\right):=\inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} $$ is achieved by a conformal metric $g \in\left[g_0\right]$, which in particular solves the $\sigma_2$-Yamabe problem, assuming $Y_2\left(M,\left[g_0\right]\right)>0$. As a consequence, for any $\left(M, g_0\right)$ with $Y_1\left(M,\left[g_0\right]\right)>$ 0 and $Y_2\left(M,\left[g_0\right]\right)>0$ one has $$ \inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}}=\inf _{g \in\left[g_0\right], R_g>0, \sigma_2(g)>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} . $$ We also show that these conclusions can fail if the condition $R_g>0$ is removed.

math.DG↗
arXiv2026

A Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature

In this paper we introduce the following Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature $R$: Find a conformal metric $g$ in a given conformal class $[g_0]$ with \[ Q_g/R_g=const. \] When the dimension $n\ge 5$, we first prove a new Sobolev inequality between the total $Q$-curvature and the total scalar curvature on $\mathbb{S}^n$ ($n\ge 5$), namely \[\frac{\int_{\mathbb{S}^n} Q_g d v_g}{\left(\int_{\mathbb{S}^n} R_g d v_g\right)^{\frac{n-4}{n-2}}} \geq \frac{\int_{\mathbb{S}^n} Q_{g_{\mathbb{S}^n}} d v\left(g_{\mathbb{S}^n}\right)}{\left(\int_{\mathbb{S}^n} R_{g_{\mathbb{S}^n}} d v\left(g_{\mathbb{S}^n}\right)\right)^{\frac{n-4}{n-2}}}\] for any $g$ in the conformal class of the round metric $g_{\mathbb{S}^n}$ with positive scalar curvature, with equality if and only if $g$ is also a metric with constant sectional curvature. With this inequality we introduce a new Yamabe constant $Y_{4,2}(M,[g_0])$ and prove the existence of the above problem provided that $Y_{4,2}(M,[g_0]) <Y_{4,2} (\mathbb{S}^n, [g_{\mathbb{S}^n}]).$ This strict inequality is proved if $(M,g)$ is not conformally equivalent to the round sphere. This follows from a crucial relation between $Y_{4,2}$ and the ordinary Yamabe constant $Y(M,[g_0])$, $Y_{4,2} (M, [g_0]) \le c(n) Y(M, [g_0])^{\frac n{n-2}}$ with equality if and only if $(M, g_0)$ is conformally equivalent to an Einstein manifold. Finally, we prove that on a closed $n$-dimensional Riemannian manifold $(M,g_{0})$ with semi-positive $Q$-curvature and non-negative scalar curvature, the above Yamabe problem is solvable, thanks to the maximum principle of Gursky-Malchiodi [33]. The proof for $n=3$ and $n=4$ follows closely the methods developed by Hang-Yang in [40], Gursky-Malchiodi in [33], and Chang-Yang in [12].

math.DG↗
arXiv2025

A Generalization of Caffarelli's Contraction Theorem to Nearly Spherical Manifolds

We show that every nearly spherical manifold can be realized as the volume-preserving image of a round sphere, via the Brenier-McCann optimal transport map. This theorem extends Caffarelli's contraction theorem to nearly spherical manifolds and yields, as a corollary, a proof of a perturbative form of Milman's conjecture. The proof is based on a novel stability result for optimal transport maps on the sphere.

math.AP↗
arXiv2025

On the problem of filling by a Poincar\'e-Einstein metric in dimension 4

Given a metric defined on a manifold of dimension three, we study the problem of finding a conformal filling by a Poincar\'e-Einstein metric on a manifold of dimension four. We establish a compactness result for classes of conformally compact Einstein $4$-manifolds under conformally invariant conditions. A key step in the proof is a result of rigidity for the hyperbolic metric on $\mathbb {B}^4$ or $ S^1 \times \mathbb{B}^3$. As an application, we also derive some existence results of conformal filling in for metrics in a definite size neighborhood of the canonical metric; when the conformal infinity is either $S^3$ or $S^1 \times S^2$.

math.DG↗
arXiv2025

Long-Tailed Out-of-Distribution Detection with Refined Separate Class Learning

Out-of-distribution (OOD) detection is crucial for deploying robust machine learning models. However, when training data follows a long-tailed distribution, the model's ability to accurately detect OOD samples is significantly compromised, due to the confusion between OOD samples and head/tail classes. To distinguish OOD samples from both head and tail classes, the separate class learning (SCL) approach has emerged as a promising solution, which separately conduct head-specific and tail-specific class learning. To this end, we examine the limitations of existing works of SCL and reveal that the OOD detection performance is notably influenced by the use of static scaling temperature value and the presence of uninformative outliers. To mitigate these limitations, we propose a novel approach termed Refined Separate Class Learning (RSCL), which leverages dynamic class-wise temperature adjustment to modulate the temperature parameter for each in-distribution class and informative outlier mining to identify diverse types of outliers based on their affinity with head and tail classes. Extensive experiments demonstrate that RSCL achieves superior OOD detection performance while improving the classification accuracy on in-distribution data.

cs.LG↗
arXiv2023

READ: Aggregating Reconstruction Error into Out-of-distribution Detection

Detecting out-of-distribution (OOD) samples is crucial to the safe deployment of a classifier in the real world. However, deep neural networks are known to be overconfident for abnormal data. Existing works directly design score function by mining the inconsistency from classifier for in-distribution (ID) and OOD. In this paper, we further complement this inconsistency with reconstruction error, based on the assumption that an autoencoder trained on ID data can not reconstruct OOD as well as ID. We propose a novel method, READ (Reconstruction Error Aggregated Detector), to unify inconsistencies from classifier and autoencoder. Specifically, the reconstruction error of raw pixels is transformed to latent space of classifier. We show that the transformed reconstruction error bridges the semantic gap and inherits detection performance from the original. Moreover, we propose an adjustment strategy to alleviate the overconfidence problem of autoencoder according to a fine-grained characterization of OOD data. Under two scenarios of pre-training and retraining, we respectively present two variants of our method, namely READ-MD (Mahalanobis Distance) only based on pre-trained classifier and READ-ED (Euclidean Distance) which retrains the classifier. Our methods do not require access to test time OOD data for fine-tuning hyperparameters. Finally, we demonstrate the effectiveness of the proposed methods through extensive comparisons with state-of-the-art OOD detection algorithms. On a CIFAR-10 pre-trained WideResNet, our method reduces the average FPR@95TPR by up to 9.8% compared with previous state-of-the-art.

cs.CV↗
arXiv2021

On compactness conformally compact Einstein manifolds and uniqueness of Graham-Lee metrics, III

In this paper, we establish a compactness result for a class of conformally compact Einstein metrics defined on manifolds of dimension $d\ge 4$. As an application, we derive the global uniqueness of a class of conformally compact Einstein metric defined on the $d$-dimensional ball constructed in the earlier work of Graham-Lee with $d\ge 4$. As a second application, we establish some gap phenomenon for a class of conformal invariants.

math.DG↗
arXiv2019

Compactness of conformally compact Einstein 4-manifolds II

In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as applications, we derive some compactness results under perturbation conditions when the L^2-norm of the Weyl curvature is small. We also derive the global uniqueness of conformally compact Einstein metrics on the 4-Ball constructed in the earlier work of Graham-Lee.

math.DG↗
arXiv2018

Compactness of conformally compact Einstein manifolds in dimension 4

In this paper, we establish some compactness results of conformally compact Einstein metrics on $4$-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology of the manifolds.

math.DG↗
arXiv2018

A remark on compact H-surfaces into R^3

Using the functional associated with the optimal Wente inequality for pairs of functions on 2-dimensional domains, we show the existence of solutions to the H-system on an annulus satisfying Neumann boundary conditions.

math.AP↗
arXiv2013

A penrose inequality for graphs over Kottler space

In this work, we prove an optimal Penrose inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Kottler space. Such inequality relies heavily on an optimal weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Kottler space.

math.DG↗
arXiv2013

On lattices with finite Coulombian interaction energy in the plane

Given a discrete set $Λ$ in the plane (we will also say a {\em lattice}) and a real number $m\ge 0$, the renormalized energy introduced in \cite{ss1} heuristically describes the interaction energy of unit charges placed at the points of $Λ$ with a uniform negative background of density $m\in\R$. It is defined in several steps, following mostly \cite{ss1}.

math-ph↗
arXiv2013

The GBC mass for asymptotically hyperbolic manifolds

The paper consists of two parts. In the first part, by using the Gauss-Bonnet curvature, which is a natural generalization of the scalar curvature, we introduce a higher order mass, the Gauss-Bonnet-Chern mass $m^{\H}_k$, for asymptotically hyperbolic manifolds and show that it is a geometric invariant. Moreover, we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding Penrose type inequality for this mass and weighted Alexandrov-Fenchel inequalities in the hyperbolic space $\H^n$. In the second part, we establish these weighted Alexandrov-Fenchel inequalities in $\H^n$ for any horospherical convex hypersurface $Σ$. As an application, we obtain an optimal Penrose type inequality for the new mass defined in the first part for asymptotically hyperbolic graphs with a horizon type boundary $Σ$, provided that a dominant energy condition $\tilde L_k\ge0$ holds. Both inequalities are optimal.

math.DG↗
arXiv2013

A new mass for asymptotically flat manifolds

In this paper we introduce a mass for asymptotically flat manifolds by using the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is a geometric invariant, if the Gauss-Bonnet curvature is integrable and the decay order $τ$ satisfies $τ> \frac {n-4}{3}.$ Then we show a positive mass theorem for asymptotically flat graphs over ${\mathbb R}^n$. Moreover we obtain also Penrose type inequalities in this case.

math.DG↗
arXiv2013

A positive mass theorem in the Einstein-Gauss-Bonnet theory

As an interesting application of the Einstein-Gauss-Bonnet theory and our work on the Gauss-Bonnet-Chern mass (Ge, Wang, Wu), we obtain a positive mass theorem for asymptotically flat graphs in $\R^{n+1}$ under a condition that $R+αL_2$ is non-negative, where $R$ is the scalar curvature, $α\in\R$ a constant and $L_2$ the second Gauss-Bonnet curvature. A Penrose type inequality is also obtained in the case $α>0$.

math.DG↗
arXiv2013

Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II

In this paper we first establish an optimal Sobolev type inequality for hypersurfaces in $\H^n$(see Theorem \ref{mainthm1}). As an application we obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and quermassintegrals. Precisely, we prove a following geometric inequality in the hyperbolic space $\H^n$, which is a hyperbolic Alexandrov-Fenchel inequality, \begin{equation*} \begin{array}{rcl} \ds \int_Σ\s_{2k}\ge \ds\vs C_{n-1}^{2k}ω_{n-1}\left\{\left(\frac{|Σ|}{ω_{n-1}} \right)^\frac 1k + \left(\frac{|Σ|}{ω_{n-1}} \right)^{\frac 1k\frac {n-1-2k}{n-1}} \right\}^k, \end{array} \end{equation*} provided that $Σ$ is a horospherical convex, where $2k\leq n-1$. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$. Here $σ_{j}=\s_{j}(κ)$ is the $j$-th mean curvature and $κ=(κ_1,κ_2,\cdots, κ_{n-1})$ is the set of the principal curvatures of $Σ$. Also, an optimal inequality for quermassintegrals in $\H^n$ is as following: $$ W_{2k+1}(Ω)\geq\frac {ω_{n-1}}{n}\sum_{i=0}^k\frac{n-1-2k}{n-1-2k+2i}\,C_k^i\bigg(\frac{nW_1(Ω)}{ω_{n-1}}\bigg)^{\frac{n-1-2k+2i}{n-1}}, $$ provided that $Ω\subset\H^n$ is a domain with $Σ=\partialΩ$ horospherical convex, where $2k\leq n-1$. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$. Here $W_r(Ω)$ is quermassintegrals in integral geometry.

math.DG↗