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Zetian Yan

Publications and source records attributed to Zetian Yan.

At least 19 recordsLinked to original sources
arXiv2026

Sharp improved Sobolev inequalities on $\mathbb{S}^n$

We prove sharp improved Sobolev inequalities on $\mathbb{S}^n$, $n\ge 4$, for functions whose critical density $|u|^{2n/(n-2)}$ has zero barycenter, without any symmetry assumption. The Dirichlet coefficient improves by the optimal factor $2^{-2/n}$, while the coefficient of the $L^2$ term remains one. Equality holds precisely for constant functions. This establishes the sharp form of Aubin's improved Sobolev inequality in every dimension $n\ge4$.

math.AP↗
arXiv2026

Prüfer $2$-group and Milnor's Conjecture on Fundamental Groups

We construct complete, one-ended Riemannian manifolds in dimensions four and five having strictly positive Ricci curvature, with fundamental group isomorphic to the Prüfer $2$-group $C_{2^\infty}$ and any subgroup $Γ\leq \mathbb Q/\mathbb Z$, respectively. This gives counterexamples to Milnor's conjecture in the two remaining dimensions.

math.DG↗
arXiv2026

Sharp stability for mixed bubbles on hyperbolic space

We prove sharp residual stability estimates for the critical Poincaré-Sobolev equation near sums of one positive hyperbolic solution and one projected Euclidean bubble in dimensions at least four. The estimates are uniform in the relative positions of the centers. The sharp rate is linear except in dimension six, where it contains a square root of a logarithm. For nonnegative functions, the same estimates hold under an energy constraint near the sum of the two profile energies for \(λ\) outside a finite set, and an additional mass constraint removes this restriction. For arbitrary finite numbers of profiles, we identify and realize all nonnegative Palais-Smale levels. We characterize the parameters at which different pairs of profile numbers have the same energy and show that the limiting energy and mass determine both profile numbers.

math.AP↗
arXiv2026

On Dolbeault's conjecture

We prove sharp improved Sobolev inequalities on $\mathbb S^n$, $n\ge3$, for antipodally even functions and for axially symmetric functions whose critical density $|u|^{2n/(n-2)}$ has zero barycenter. In both cases, the Dirichlet coefficient improves by the optimal factor $2^{-2/n}$, while the coefficient of the $L^2$ term remains one. Equality holds precisely for constant functions. The even case resolves a conjecture of Dolbeault (2021).

math.AP↗
arXiv2026

Uniqueness of the asymptotic limits for Ricci-flat manifolds with linear volume growth

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 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--Nordström. In dimension four, these assumptions hold automatically, yielding unconditional uniqueness and convergence thus our result strengthens those of Chen--Chen, who proved exponential convergence to its asymptotic limit space for ALH instantons. Compared to cones, the greatest challenge is to bound the multiplicity two zero modes and oscillatory negative eigenvalue modes, which are new phenomena for cylinders. To overcome the new challenge, we develop a new technical tool, the constant mean curvature foliation.

math.DG↗
arXiv2026

Nonnegative Ricci Curvature and Uniformly Convex Boundary Forces Compactness

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.

math.DG↗
arXiv2026

Topology of 3-manifolds with nonnegative scalar curvature and positive harmonic functions

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.

math.DG↗
arXiv2026

Uniqueness of the asymptotic limits for Ricci-flat manifolds with linear volume growth II

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.

math.DG↗
arXiv2025

Sharp Fractional Sobolev Embeddings on Closed Manifolds

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.

math.AP↗
arXiv2025

Rigidity of CMC hypersurfaces in 5-and 6-manifolds

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.

math.DG↗
arXiv2025

Energy reduction for Fourth order Willmore energy

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}.

math.DG↗
arXiv2024

Juhl type formulas for curved Ovsienko--Redou operators

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.

math.DG↗
arXiv2023

Convergence of the Weighted Yamabe Flow

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 $n\geqslant 3$.

math.DG↗
arXiv2023

Sharp weighted CR trace Sobolev inequalities

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.

math.AP↗
arXiv2023

Improved Sobolev inequalities on CR shpere

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.

math.DG↗
arXiv2022

Some results on the weighted Yamabe problem with or without boundary

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.

math.DG↗