arXiv subjects
Publications and source records attributed to Zetian Yan.
We construct complete, one-ended Riemannian manifolds in dimensions four and five having strictly positive Ricci curvature, with fundamental group isomorphic to the Pr\"ufer $2$-group $C_{2^\infty}$. This gives counterexamples to Milnor's conjecture in the two remaining dimensions.
We confirm a compactness conjecture of M. Li. If a complete Riemannian manifold has nonnegative Ricci curvature and uniformly convex boundary in the sense that the second fundamental form satisfies $h\ge1$. Then we prove it is compact, and consequently has finite fundamental group. The proof uses monotone quantities constructed via positive proper harmonic functions with Neumann condition.
We study complete $3$-manifolds with nonnegative scalar curvature under additional regularity assumptions. We prove that a contractible such manifold is diffeomorphic to $\mathbb{R}^3$, and that an open handlebody admitting such a metric must have genus at most $1$. The proof uses exhaustions by level sets of harmonic functions and refined average gradient estimates.
We relate the uniqueness of asymptotic limits for noncollapsed Ricci flat manifolds with linear volume growth to the existence of a harmonic function asymptotic to a Busemann function. Parallel to the work of Colding--Minicozzi in the Euclidean volume growth setting, we prove uniqueness of the asymptotic limit and establish a quantitative polynomial convergence rate via a monotone quantity associated with this harmonic function, assuming such harmonic function exists and one asymptotic limit is smooth. Conversely, for an open manifold with nonnegative Ricci curvature, we show that uniqueness of the asymptotic limit implies the existence of the desired harmonic function, without assuming smoothness of the cross section.
We develop an intrinsic, heat-kernel based fractional Sobolev framework on closed Riemannian manifolds and study the critical fractional Sobolev embedding. We determine the optimal coefficient of the lower-order $L^{p}$ term and prove that the fully sharp $p$-power inequality cannot hold globally in the superquadratic range. We further establish an almost sharp inequality whose leading constant is arbitrarily close to the Euclidean best constant, and we derive improved inequalities under finitely many orthogonality constraints with respect to sign-changing test families.
Under natural assumptions on curvature and cross section, we establish the uniqueness of asymptotic limits and the exponential convergence rate for complete noncollapsed Ricci-flat manifolds with linear volume growth, which are known to only admit cylindrical asymptotic limits. In dimension four, these assumptions hold automatically, yielding unconditional uniqueness and convergence. In particular, our results show that all asymptotically cylindrical Calabi--Yau manifolds converge exponentially to their asymptotic limits, thereby answering affirmatively a question by Haskins--Hein--Nordstr\"om. In dimension four our result strengthens those of Chen--Chen, who proved exponential convergence to its asymptotic limit space for any ALH instanton.
We introduce a fourth-order Willmore-type problem for closed four-dimensional submanifolds immersed in $\mathbb{R}^n$ and establish a connected sum energy reduction for the general fourth-order Willmore energy, analogous to the seminal result of Bauer and Kuwert \cite{Bauer-Kuwert03}.
We prove Juhl type formulas for the curved Ovsienko--Redou operators and their linear analogues, which indicate the associated formal self-adjointness, thereby confirming two conjectures of Case, Lin, and Yuan. We also offer an extension of Juhl's original formula for the GJMS operators.
We prove that the curved Ovsienko--Redou operators and a related family of differential operators are formally self-adjoint. This verifies two conjectures of Case, Lin, and Yuan.
We prove that nonnegative $3$-intermediate Ricci curvature combined with uniformly positive $k$-triRic curvature implies rigidity of complete noncompact two-sided stable minimal hypersurfaces in a Riemannian manifold $(X^5,g)$ with bounded geometry. The stonger assumption of nonnegative $3$-intermediate Ricci curvature can be replaced by the nonnegativity of Ricci and biRic curvature. In particular, there is no complete noncompact stable minimal hypersurface in a closed $5$-dimensional manifold with positive sectional curvature. This extends result of Chodosh-Li-Stryker [J. Eur. Math. Soc (2025)] to $5$-dimension. We also establish rigidity results on CMC hypersurfaces with nonzero mean curvature in $5$- and $6$-manifolds.
We prove that a stable $C^{1,1}$-to-edge properly embedded free boundary minimal hypersurface $Σ^3$ of a $4$-dimensional wedge domain $Ω^4_θ$ with angle $θ\in (0,π]$ is flat.
We introduce the weighted Yamabe flow $\frac{\partial g}{\partial t}=(r^m_ϕ-R^m_ϕ)g$, $\frac{\partial ϕ}{\partial t}=\frac{m}{2}(R^m_ϕ-r^m_ϕ)$ on a smooth metric measure space $(M^n, g, e^{-ϕ}{\rm dvol}_g, m)$, where $R^m_ϕ$ denotes the associated weighted scalar curvature, and $r^m_ϕ$ denotes the mean value of the weighted scalar curvature. We prove long-time existence and convergence of the weighted Yamabe flow if the dimension $n$ satisfies $n\geqslant 3$.
We establish a sharp Sobolev trace inequality on the Siegel domain $Ω_{n+1}$ involving the weighted norm-$W^{2,2}(Ω_{n+1}, ρ^{1-2[γ]})$. The inequality is closely related the realization of fractional powers of the sub-Laplacian on the Heisenberg group $H^n=\partial Ω_{n+1}$ as generalized Dirichlet-to-Neumann operators associated to the weighted poly-sublaplacian, generalizing observations of Frank--González--Monticelli--Tan.
We establish improved CR Sobolev inequalities on CR sphere under the vanishing of higher order moments of the volume element. As a direct application, we give a simpler proof of the existence and the classification of minimizers of the CR invariant Sobolev inequalities which avoids complicated computation in Frank and Lieb's proof. Our argument relies on nice commutator identities involving the CR intertwining operators on CR sphere and handles both the fractional and integral cases. In the same spirit, we derive the classical sharp Sobolev inequalities using commutator identities on the sphere.
Let $(M^n,g,e^{-ϕ}dV_g,e^{-ϕ}dA_g,m)$ be a compact smooth metric measure space with boundary with $n\geqslant 3$. In this article, we consider several Yamabe-type problems on a compact smooth metric measure space with or without boundary: uniqueness problem on the weighted Yamabe problem with boundary, characterization of the weighted Yamabe solitons with boundary and the existence of positive minimizers in the weighted Escobar quotient.
The weighted Yamabe flow was the geometric flow introduced to study the weighted Yamabe problem on smooth metric measure spaces. Carlotto, Chodosh and Rubinstein have studied the convergence rate of the Yamabe flow. Inspired by their result, we study in this paper the convergence rate of the weighted Yamabe flow.
We introduce a Yamabe-type flow \begin{align*} \left\{ \begin{array}{ll} \frac{\partial g}{\partial t} &=(r^m_ϕ-R^m_ϕ)g \\ \frac{\partial ϕ}{\partial t} &=\frac{m}{2}(R^m_ϕ-r^m_ϕ) \end{array} \right. ~~\mbox{ in }M ~~\mbox{ and }~~ H^m_ϕ=0 ~~\mbox{ on }\partial M \end{align*} on a smooth metric measure space with boundary $(M,g, v^mdV_g,v^mdA_g,m)$, where $R^m_ϕ$ is the associated weighted scalar curvature, $r^m_ϕ$ is the average of the weighted scalar curvature, and $H^m_ϕ$ is the weighted mean curvature. We prove the long-time existence and convergence of this flow.
We improve higher-order CR Sobolev inequalities on $S^{2n+1}$ under the vanishing of higher order moments of the volume element. As an application, we give a new and direct proof of the classification of minimizers of the CR invariant higher-order Sobolev inequalities. In the same spirit, we prove almost sharp Sobolev inequalities for GJMS operators to general CR manifolds, and obtain the existence of minimizers in $C^{2k}(N)$ of higher-order CR Yamabe-type problems when $Y_k(N)<Y_k(\mathbb{H}^n)$.