Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.19667

Remarks on prescribed scalar curvature functions

Mingxiang Li·Quoc Anh Ngo·Xingwang Xu
View original source ↗Read source document ↗Export metadata ↓

Abstract

This short article is devoted to the study of the positive solution of Yamabe-type semi-linear equation with prescribed coefficient on general closed manifolds. Our main result is to provide some sufficient conditions for the existence of the positive solutions and give some examples to show that those conditions are sharp. An application to scalar-flat manifold settles a long standing question in prescribing scalar curvature problem in negative form.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mingxiang Li, Quoc Anh Ngo, Xingwang Xu. 2026-10-14. Remarks on prescribed scalar curvature functions. https://arxiv.org/abs/2610.19667

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Lin-Lu-Yau Ricci curvature on hypergraphs

In this paper, we introduce a unified framework for defining Lin-Lu-Yau (LLY) Ricci curvature on both undirected and directed hypergraphs. By establishing upper bounds and monotonicity properties for the parameterized curvature $κ_α$, we justify the well-posedness and compatibility of our definitions. Furthermore, we prove a Bonnet-Myers-type theorem for hypergraphs, which highlights the potential of LLY Ricci curvature in hypergraph analysis, particularly in studying geometric and structural properties. Our results extends the foundational definitions of graph Ricci curvature by Ollivier and Lin-Lu-Yau to the hypergraph setting.

math.DG↗
arXiv2026

Characterizations of weak almost ${\mathcal S}$-manifolds with curvature properties

Weak metric structures, introduced by Rovenski and Wolak in 2022, extend Yano's $f$-structure and almost contact metric structure. In this paper, we investigate curvature phenomena of weak almost ${\cal S}$-manifolds (w.a.$\,\cal S$-manifolds) focusing on the $f$-$(κ,μ)$-nullity condition and its special case $R_{X,Y}\,ξ=0$. We establish several results that generalize known rigidity theorems for almost ${\cal S}$-manifolds. First, using the partial Ricci flow, we obtain dynamical characterizations of $\cal S$-manifolds: starting from a w.a.$\,\cal S$-structure satisfying the curvature condition of $\cal S$-manifolds or the $f$-$(1,μ)$-nullity condition, the flow evolves the structure exponentially fast toward an $\cal S$-structure. This extends results of Cappelletti Montano and Di~Terlizzi to the weak metric setting. Next, we identify conditions under which a w.a.\,${\cal S}$-manifold admits a bi-Legendrian structure with totally geodesic foliations. Finally, for w.a.$\,\cal S$-manifolds with $κ=μ=0$, we prove a splitting theorem in which one factor is flat and the another has positive constant sectional curvature, generalizing classical results for almost $\cal S$-geometry. These findings have consequences for the theory of Sasakian and $\cal S$- manifolds, the geometry of bi-Legendrian structures, and the behavior of weak metric contact manifolds under curvature constraints.

math.DG↗
arXiv2026

Kähler geometry on total spaces of vector bundles over elliptic curves

We study function theory and Kähler geometry on total spaces of vector bundles over an elliptic curve. Our main result characterizes the vector bundles whose total spaces admit complete Kähler metrics with nonnegative bisectional curvature ($BI \geq 0$ for short). We also obtain explicit expressions for all complete Kähler metrics with $BI \geq 0$ on the total space of a line bundle over a complex torus. These results are related to a general conjecture concerning the structure of complete Kähler manifolds with $BI \geq 0$. For rank-two vector bundles of degree zero over elliptic curves, we study holomorphic functions and automorphisms on the corresponding total spaces, and construct complete Gauduchon Hermitian metrics with vanishing Chern-Ricci curvature. These metrics are natural in the sense that the associated spaces of holomorphic functions of polynomial growth coincide with the ``polynomials'' on these total spaces.

math.DG↗