arXiv subjects

Haoxuan Cheng

Publications and source records attributed to Haoxuan Cheng.

7 recordsLinked to original sources

arXiv2026

Unbounded normalized scalar curvature integrals in dimension four

We give counterexamples to Yau's question on normalized scalar curvature integrals in every dimension $n\geq4$. For each such $n$, there exists a smooth complete Riemannian metric $g$ on $\mathbb{R}^n$ with nonnegative Ricci curvature and a pole such that \[ \lim_{R\to\infty}R^{2-n}\int_{B_q(R)}\mathrm{Scal}_g\,\mathrm dV_g=+\infty \] for every fixed $q\in\mathbb{R}^n$. Here $B_q(R)$ denotes the geodesic ball of radius $R$ centered at $q$, and $\mathrm{Scal}_g$ is the scalar curvature of $g$.

math.DG↗

arXiv2026

Singular Rotational Self-Similar Tori for Odd $\sigma_k$-Curvature Flows

For every pair of integers $3\leq k<n$ with $k$ odd, we construct a compact embedded rotational torus in $\mathbb{R}^{n+1}$ whose homothetic dilations satisfy the unnormalised $\sigma_k$-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has H\"older regularity $C^{1,1/k}$ and Sobolev regularity $W^{2,p}$ for every $1\leq p<k/(k-1)$. Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation $\langle X,\nu\rangle=-\sigma_k$, where $X$ is the position vector and $\nu$ is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical $C^2$ rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.

math.DG↗

arXiv2026

Volume Growth and Recurrence of Fractional Powers of the Laplace--Beltrami Operator

Let $M$ be a connected geodesically complete Riemannian manifold without boundary, write $\mu$ for Riemannian volume, and set $V(o,r)=\mu(B(o,r))$ for geodesic balls centered at $o$. For $0<\alpha<2$, let $X^{(\alpha)}$ be the process obtained by subordinating Brownian motion with an independent $\alpha/2$-stable subordinator; its $L^2$-generator is $-(-\Delta)^{\alpha/2}$. We prove that \[ \int^\infty\frac{dt}{V(o,t^{1/\alpha})}=\infty \] implies that $X^{(\alpha)}$ is recurrent. The proof uses a spectral trace estimate and radial cutoffs on $M\times(0,\infty)$, where the auxiliary measure is $y^{1-\alpha}\,d\mu\,dy$. It proves the sufficient implication in Grigor'yan's Problem~26.

math.PR↗

arXiv2026

Edge Subdivision and the Perron Eigenvalue of Tree Ricci Matrices

The Ricci matrix $R_T$ of a finite tree encodes its discrete Einstein metrics via the Perron eigenvector, with Lin-Lu-Yau's Ollivier Ricci curvature: $\kappa = -\lambda_{\max}(R_T)$. We show that edge subdivision, the natural operation of lengthening a tree, can decrease, preserve, or increase $\lambda_{\max}$. Compressing each branch into a scalar feedback function via the Schur complement reduces the spectral problem to a one-dimensional Chebyshev equation. We obtain an exact one-step trichotomy, a scalar transmission equation for arbitrary length, and the long-chain limit. Examples on double stars, including an asymmetric case where subdivision strictly increases $\lambda_{\max}$, illustrate the theory.

math.DG↗

arXiv2026

Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees

Let $R_T$ be the Ricci matrix of a finite tree $T$ introduced in \cite{BaiChengHua2026}, the largest eigenvalue $λ_{\max}(R_T)$ determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the asymptotic behavior of the sequence $λ_k = λ_{\max}(R_{T_k})$ obtained by repeatedly adding pendant edges at a fixed vertex. We prove that $λ_k$ converges to a limit $λ_\infty$ that depends only on the local branch data of $T$, and establish a first-order asymptotic expansion: \[ λ_k = λ_\infty + \fracα{d+k} + O\!\left(\frac{1}{(d+k)^2}\right), \] where $d$ is the degree of the original vertex, and the coefficient $α$ is given by a spectral projection. As a corollary, when $α\neq 0$, $λ_k$ is eventually strictly monotonic (increasing or decreasing). This theory reveals the fine influence of local leaf addition on the global spectrum.

math.DG↗

arXiv2026

Positive-Curvature Discrete Einstein Metrics on Trees

For a weighted tree, the Lin--Lu--Yau Ricci curvature admits an explicit formula in terms of the edge weights. Consequently, the constant-curvature equation is equivalent to an eigenvalue problem for an edge-indexed Ricci matrix $R_T$. Building on the spectral characterization of discrete Einstein metrics on trees, we classify all finite trees whose discrete Einstein metric has positive curvature, equivalently all trees satisfying $\lambda_{\max}(R_T)<0$. For caterpillars with spine order $m\ge 12$, this occurs precisely for the endpoint families $T_m(a,0,\ldots,0,b)$ with $1\le a,b\le 3$ and $(a,b)\ne(3,3)$. The remaining cases $3\le m\le 11$ are settled by an exact finite verification using rational characteristic polynomials and Sturm root counts. We also determine the zero level set $\lambda_{\max}(R_T)=0$: among caterpillars, it consists of the stable family $(3,0,\ldots,0,3)$ together with nine exceptional short-spine caterpillars, while $S_3^2$ is the unique non-caterpillar zero example.

math.DG↗

arXiv2026

Discrete Einstein metrics on trees

We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. We establish a sharp upper bound for the largest eigenvalue of the associated Ricci matrix in terms of the maximum degree. Turning to structural properties, notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge.

math.DG↗