arXiv subjects

Xingzhe Li

Publications and source records attributed to Xingzhe Li.

8 recordsLinked to original sources

arXiv2026

Infinite existence of equivariant minimal hypersurfaces

For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show that there are infinitely many $G$-invariant minimal hypersurfaces. Under the assumption that $M$ contains at most a finite number of minimal $G$-hypersurfaces admitting no $G$-invariant unit normal, we further show that each $G$-homology class of $M$ admits infinitely many distinct realizations by embedded minimal $G$-hypersurfaces. The proof relies on a new algorithm that employs multi-stage maximal cuttings. As part of this work, we also established an equivariant min-max theory in manifolds with cylindrical ends.

math.DG↗

arXiv2025

Area Rigidity for the Regular Representation of Surface Groups

Let $\tildeΣ$ be the universal cover of a closed surface $Σ$ of genus at least $2$. We characterize all equivariantly area-minimizing maps from $\tildeΣ$ to a Hilbert sphere, which are equivariant with respect to an isometric action of $π_1(Σ)$ weakly equivalent to the regular representation. As part of our proof, we classify all minimal surfaces in Hilbert spheres with constant negative Gaussian curvature. This builds on earlier results of E. Calabi, K. Kenmotsu, R. Bryant.

math.DG↗

arXiv2025

What to Retrieve for Effective Retrieval-Augmented Code Generation? An Empirical Study and Beyond

Repository-level code generation remains challenging due to complex code dependencies and the limitations of large language models (LLMs) in processing long contexts. While retrieval-augmented generation (RAG) frameworks are widely adopted, the effectiveness of different retrieved information sources-contextual code, APIs, and similar snippets-has not been rigorously analyzed. Through an empirical study on two benchmarks, we demonstrate that in-context code and potential API information significantly enhance LLM performance, whereas retrieved similar code often introduces noise, degrading results by up to 15%. Based on the preliminary results, we propose AllianceCoder, a novel context-integrated method that employs chain-of-thought prompting to decompose user queries into implementation steps and retrieves APIs via semantic description matching. Through extensive experiments on CoderEval and RepoExec, AllianceCoder achieves state-of-the-art performance, improving Pass@1 by up to 20% over existing approaches.

cs.SE↗

arXiv2024

Creating a Cooperative AI Policymaking Platform through Open Source Collaboration

Advances in artificial intelligence (AI) present significant risks and opportunities, requiring improved governance to mitigate societal harms and promote equitable benefits. Current incentive structures and regulatory delays may hinder responsible AI development and deployment, particularly in light of the transformative potential of large language models (LLMs). To address these challenges, we propose developing the following three contributions: (1) a large multimodal text and economic-timeseries foundation model that integrates economic and natural language policy data for enhanced forecasting and decision-making, (2) algorithmic mechanisms for eliciting diverse and representative perspectives, enabling the creation of data-driven public policy recommendations, and (3) an AI-driven web platform for supporting transparent, inclusive, and data-driven policymaking.

cs.CY↗

arXiv2024

Existence of embedded minimal tori in three-spheres with positive Ricci curvature

In this paper, we prove the strong Morse inequalities for the area functional in the space of embedded tori and spheres in the three sphere. As a consequence, we prove that in the three dimensional sphere with positive Ricci curvature, there exist at least 4 distinct embedded minimal tori. Suppose in addition that the metric is bumpy, then the three-sphere contains at least 9 distinct embedded minimal tori. The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory proved by the second author and X. Zhou.

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

Generic Scarring for Minimal Hypersurfaces in Manifolds Thick at Infinity with a Thin Foliation at Infinity

We show generic scarring phenomenon for minimal hypersurfaces in a class of complete non-compact manifolds. In particular, we prove that for any metric $g$ in a $C^{\infty}$-generic subset of the family of complete metrics which are thick at infinity with a thin foliation at infinity on a fixed $M^{n+1}$ of dimension $3 \leq (n + 1) \leq 7$, to any connected, closed, embedded, $2$-sided, stable minimal hypersurface $S \subset (M, g)$, there exists a sequence of closed, embedded, minimal hypersurfaces $\{Σ_{k}\}$ scarring along $S$, in the sense that the area of $Σ_{k}$ diverges to infinity, and when properly renormalized, $Σ_{k}$ converges to $S$ as varifolds.

math.DG↗

arXiv2021

Homological Filling Functions with Coefficients

How hard is it to fill a loop in a Cayley graph with an unoriented surface? Following a comment of Gromov in "Asymptotic invariants of infinite groups", we define homological filling functions of groups with coefficients in a group $R$. Our main theorem is that the coefficients make a difference. That is, for every $n \geq 1$ and every pair of coefficient groups $A, B \in \{\mathbb{Z},\mathbb{Q}\} \cup \{\mathbb{Z}/p\mathbb{Z} : p\text{ prime}\}$, there is a group whose filling functions for $n$-cycles with coefficients in $A$ and $B$ have different asymptotic behavior.

math.GR↗