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Weike Yu

Publications and source records attributed to Weike Yu.

15 recordsLinked to original sources
arXiv2026

$L^{1}$-Integrability of $L^{2}$-Harmonic Forms and the Hopf Conjecture

In this note, we study $L^{2}$-harmonic forms on complete simply-connected Riemannian manifolds with non-positive sectional curvature. We first establish an a priori $L^{\infty}$-estimate for such forms via Moser iteration, under the curvature bounds $-K\leq\mathrm{sec}_{g}\leq0$. We then prove that any $L^{2}$-harmonic form which is also $L^{1}$-integrable must vanish identically. Consequently, on the universal cover of a closed non-positively curved manifold, the $k$-th $L^{2}$-Betti number vanishes if and only if every $L^{2}$-harmonic $k$-form is $L^{1}$-integrable. This criterion reformulates a topological vanishing statement as an analytic integrability condition.

math.DG↗
arXiv2025

Prescribed Chern scalar curvatures on complete Hermitian manifolds

In this paper, we investigate the problem of prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds, and generalize the Aviles-McOwen's existence results [J. Differential Geom., 21 (1985): 269-281] from Poincar\'e disks to higher dimensional Hermitian manifolds.

math.DG↗
arXiv2025

Prescribed Chern scalar curvature flow on compact Hermitian manifolds with negative Gauduchon degree

In this paper, we present a unified flow approach to prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree. When the conformal class of its Hermitian metric contains a balanced metric, we give some sufficient conditions on the candidate curvature function $f$ which guaranties the convergence of the flow to a conformal Hermitian metric whose Chern scalar curvature is $f$.

math.DG↗
arXiv2024

Liouville theorem for $V$-harmonic heat flows

In this paper, we investigate $V$-harmonic heat flows from complete Riemannian manifolds with nonnegative Bakry-Emery Ricci curvature to complete Riemannian manifolds with sectional curvature bounded above. We give a gradient estimate of ancient solutions to this flow and establish a Liouville type theorem.

math.DG↗
arXiv2024

The Kazdan-Warner problem on compact K\"ahler surfaces

In this paper, we investigate a Kazdan-Warner problem on compact K\"ahler surfaces, which corresponds to prescribing sign-changing Chern scalar curvatures, and establish a Chen-Li type existence theorem on compact K\"ahler surfaces when the candidate curvature function is of negative average. Moreover, we give an alternative proof of Ding-Liu's theorem [Trans. Amer. Math. Soc. 347(1995) 1059-1066] on prescribing sign-changing Gaussian curvatures.

math.DG↗
arXiv2022

Prescribed Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree

In this paper, we investigate the problem of prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree. By studying the convergence of the associated geometric flow, we obtain some existence results when the candidate curvature function is nonzero and nonpositive. Furthermore, we also consider the case that the candidate curvature function is sign-changing, and establish some existence and nonexistence results.

math.DG↗
arXiv2022

A generalization of the Schwarz lemma for transversally harmonic maps

In this paper, we consider transversally harmonic maps between Riemannian manifolds with Riemannian foliations. In terms of the Bochner techniques and sub-Laplacian comparison theorem, we are able to establish a generalization of the Schwarz lemma for transversally harmonic maps of bounded generalized transversal dilatation. In addition, we also obtain a Schwarz type lemma for transversally holomorphic maps between K\"ahler foliations.

math.DG↗
arXiv2022

On subelliptic harmonic maps with potential

Let $(M,H,g_H;g)$ be a sub-Riemannian manifold and $(N,h)$ be a Riemannian manifold. For a smooth map $u: M \to N$, we consider the energy functional $E_G(u) = \frac{1}{2} \int_M[|\mathrm{d}u_H|^2-2G(u)] \mathrm{d}V_M$, where $\mathrm{d}u_H$ is the horizontal differential of $u$, $G:N\to \mathbb{R}$ is a smooth function on $N$. The critical maps of $E_G(u)$ are referred to as subelliptic harmonic maps with potential $G$. In this paper, we investigate the existence problem for subelliptic harmonic maps with potentials by a subelliptic heat flow. Assuming that the target Riemannian manifold has non-positive sectional curvature and the potential $G$ satisfies various suitable conditions, we prove some Eells-Sampson type existence results when the source manifold is either a step-$2$ sub-Riemannian manifold or a step-$r$ sub-Riemannian manifold whose sub-Riemannian structure comes from a tense Riemannian foliation.

math.DG↗
arXiv2021

Tamed exhaustion functions and Schwarz type lemmas for almost Hermitian manifolds

In this paper, we study a special exhaustion function on almost Hermitian manifolds and establish the existence result by using the Hessian comparison theorem. From the viewpoint of the exhaustion function, we establish related Schwarz type lemmas for almost holomorphic maps between two almost Hermitian manifolds. As corollaries, we deduce Liouville type theorems for almost holomorphic maps.

math.CV↗
arXiv2021

Prescribed Webster scalar curvatures on compact pseudo-Hermitian manifolds

In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of self-adjoint operators, we can describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.

math.DG↗
arXiv2020

Generalized maximum principles and stochastic completeness for pseudo-Hermitian manifolds

In this paper, we establish a generalized maximum principle for pseudo-Hermitian manifolds. As corollaries, Omori-Yau type maximum principles for pseudo-Hermitian manifolds are deduced. Moreover, we prove that the stochastic completeness for the heat semigroup generated by the sub-Laplacian is equivalent to the validity of a weak form of the generalized maximum principles. Finally, we give some applications of these generalized maximum principles.

math.DG↗
arXiv2020

Schwarz type lemmas for generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we study CR Carath\'eodory pseudodistance on CR manifolds.

math.DG↗
arXiv2019

Schwarz type lemmas for pseudo-Hermitian manifolds

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the \emph{CR} maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and comparison theorems in the pseudo-Hermitian version, we are able to establish several Schwarz type results for both the \emph{CR} maps and the transversally holomorphic maps between pseudo-Hermitian manifolds. Finally, we also discuss the \emph{CR} hyperbolicity problem for pseudo-Hermitian manifolds.

math.DG↗