Complete Manifold Research Articles

$$L^{2}$$ L 2 harmonic forms on complete special holonomy manifolds

In this article, we consider $$L^{2}$$ harmonic forms on a complete non-compact Riemannian manifold X with a nonzero parallel form $$\omega $$ . The main result is that if $$(X,\omega )$$ is a complete $$G_{2}$$ - (or $$\textit{Spin}(7)$$ -) manifold with a d(linear) $$G_{2}$$ - (or $$\textit{Spin}(7)$$ -) structure form $$\omega $$ , then the $$L^{2}$$ harmonic 2-forms on X vanish. As an application, we prove that the instanton equation with square-integrable curvature on $$(X,\omega )$$ only has trivial solution. We would also consider the Hodge theory on the principal G-bundle E over $$(X,\omega )$$ .

A sphere theorem for Bach-flat manifolds with positive constant scalar curvature

A classic gap theorem for complete non-compact Ricci flat manifolds by M. Anderson suggests that one can get the rigidity of certain spaces simply by passing local curvature estimates on geodesic balls to global. Using similar ideas, Kim showed that a 4-dimensional complete non-compact Bach-flat manifold with vanishing scalar curvature and small L2-curvature tensor has to be flat. Unfortunately, this method does not generalize to compact manifolds without boundary. Applying a different approach, we show that a closed Bach-flat Riemannian manifold with positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either L∞ or Ln2-norm. These results generalize a rigidity theorem of positive Einstein manifolds due to M.-A. Singer. As an application, we can partially recover the well-known Chang–Gursky–Yang's 4-dimensional conformal sphere theorem.

Canonical sphere bundles of the Grassmann manifold

For a given Hilbert space $$\mathcal H$$ , consider the space of self-adjoint projections $$\mathcal P(\mathcal H)$$ . In this paper we study the differentiable structure of a canonical sphere bundle over $$\mathcal P(\mathcal H)$$ given by $$\begin{aligned} \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \Vert f\Vert =1\, \}. \end{aligned}$$ We establish the smooth action on $$\mathcal R$$ of the group of unitary operators of $$\mathcal H$$ , and it thereby turns out that the connected components of $$\mathcal R$$ are homogeneous spaces. Then we study the metric structure of $$\mathcal R$$ by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into $$\mathcal R$$ by the natural action of the unitary group. Then we study the restricted bundle $$\mathcal R_2^+$$ given by considering only the projections in the restricted Grassmannian, locally modeled by Hilbert–Schmidt operators. Therefore we endow $$\mathcal R_2^+$$ with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi–Civita connection of this metric and establish a Hopf–Rinow theorem for $$\mathcal R_2^+$$ , again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.

M-quasi-Einstein metric and contact geometry

We study m-quasi-Einstein metric in the framework of contact metric manifolds. The existence of such metric has been confirmed on the class of $$ \eta $$ -Einstein K-contact manifold, in which the potential vector field V is a constant multiple of the Reeb vector field $$\xi $$ . Next, we consider closed m-quasi-Einstein metric on a complete K-contact manifold and prove that it is Sasakian and Einstein provided $$m\ne 1$$ . We also proved that, if a K-contact manifold M admits an m-quasi-Einstein metric such that the potential vector field V is conformal, then V becomes Killing and M is $$ \eta $$ -Einstein. Finally, we obtain a couple of results on a contact metric manifold M admitting an m-quasi-Einstein metric, whose potential vector field is a point wise collinear with the Reeb vector field.

Lie groups with flat Gauduchon connections

We pursue the research line proposed in Yang and Zheng (Acta. Math. Sinica (English Series), 34(8):1259–1268, 2018) about the classification of Hermitian manifolds whose s-Gauduchon connection $$\nabla ^s =(1-\frac{s}{2})\nabla ^c + \frac{s}{2}\nabla ^b$$ is flat, where $$s\in {\mathbb {R}}$$ and $$\nabla ^c$$ and $$\nabla ^b$$ are the Chern and the Bismut connections, respectively. We focus on Lie groups equipped with a left invariant Hermitian structure. Such spaces provide an important class of Hermitian manifolds in various contexts and are often a valuable vehicle for testing new phenomena in complex and Hermitian geometry. More precisely, we consider a connected 2n-dimensional Lie group G equipped with a left-invariant complex structure J and a left-invariant compatible metric g and we assume that its connection $$\nabla ^s$$ is flat. Our main result states that if either n=2 or there exits a $$\nabla ^s$$ -parallel left invariant frame on G, then g must be Kähler. This result demonstrates rigidity properties of some complete Hermitian manifolds with $$\nabla ^s$$ -flat Hermitian metrics.

On the Essential Self-Adjointness of Singular Sub-Laplacians

We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This result holds under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.

Yamabe flow and metrics of constant scalar curvature on a complete manifold

In this paper, we give a condition on the initial metric which makes the global existence of Yamabe flow and we study the global behavior of the Yamabe flow in a complete noncompact Riemannian manifold. As application we study the ADM mass monotonicity of Yamabe flow in AF manifolds. We use the variational method to study the existence problem of metrics with constant scalar curvature on complete non-compact Riemannian manifolds and we can give a partial affirmative answer to a question posed by Kazdan (Math Ann 261(2):227–234, 1982).

On the semimartingale property of Brownian bridges on complete manifolds

I prove that every adapted Brownian bridge on a geodesically complete connected Riemannian manifold is a semimartingale including its terminal time, without any further assumptions on the geometry. In particular, it follows that every such process can be horizontally lifted to a smooth principal fiber bundle with connection, including its terminal time. The proof is based on a localized Hamilton-type gradient estimate by Arnaudon/Thalmaier.

Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds

We show vanishing theorems of $L^2$-cohomology groups of Kodaira–Nakano type on complete Hessian manifolds by introducing a new operator $\partial'_F$. We obtain further vanishing theorems of $L^2$-cohomology groups $L^2H^{p,q}_{\bar{\partial}}(\Omega)$ on a regular convex cone $\Omega$ with the Cheng–Yau metric for $p>q$.

Calabi-Yau manifolds and SU(3) structure

We show that non-trivial SU(3) structures can be constructed on large classes of Calabi-Yau three-folds. Specifically, we focus on Calabi-Yau three-folds constructed as complete intersections in products of projective spaces, although we expect similar methods to apply to other constructions and also to Calabi-Yau four-folds. Among the wide range of possible SU(3) structures we find Strominger-Hull systems, suitable for heterotic or type II string compactifications, on all complete intersection Calabi-Yau manifolds. These SU(3) structures of Strominger-Hull type have a non-vanishing and non-closed three-form flux which needs to be supported by source terms in the associated Bianchi identity. We discuss the possibility of finding such source terms and present first steps towards their explicit construction. Provided suitable sources exist, our methods lead to Calabi-Yau compactifications of string theory with a non Ricci-flat, physical metric which can be written down explicitly and in analytic form.

Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density

In this paper, we construct a helicoidal surface with a prescribed weighted mean curvature and weighted extrinsic curvature in a 3-dimensional complete manifold with a positive density function. We get a result for the minimal case. Additionally, we give examples of a helicoidal surface with a weighted mean curvature and weighted extrinsic curvature.

A criterion for quasinormality in $$\pmb {\mathbb {C}^n}$$ C n

In this article, we give a Zalcman type renormalization result for the quasinormality of a family of holomorphic functions on a domain in $$\mathbb {C}^n$$ that takes values in a complete complex Hermitian manifold.

A Liouville theorem on complete non-Kähler manifolds

In this paper, we prove a Liouville theorem for holomorphic functions on a class of complete Gauduchon manifolds. This generalizes a result of Yau for complete Kahler manifolds to the complete non-Kahler case.

Affine vector fields on Finsler manifolds

We give characterizations of affine transformations and affine vector fields in terms of the spray. By utilizing the Jacobi type equation that characterizes affine vector fields, we prove some rigidity theorems of affine vector fields on compact or forward complete non-compact Finsler manifolds with non-positive total Ricci curvature.

Strongly and weakly affine vector fields on Finsler manifolds

In this paper, we investigate the affine vector fields on both compact and forward complete Finsler manifolds. We first give definitions of the affine transformation and the affine vector field. Unexpectedly, we find two kinds of affine fields, which are named as the strongly and weakly affine vector fields. Based on these definitions, we prove some rigidity theorems of affine fields on compact and forward complete Finsler manifolds.

Non-integrated defect of meromorphic maps on Kähler manifolds

The purpose of this article is twofold. The first is to establish a truncated non-integrated defect relation for meromorphic mappings from a complete Kahler manifold quotien of a ball into a projective variety intersecting hypersurfaces in subgeneral position. We also apply it to the Gauss mapping from a closed regular submanifold of $${\mathbb {C}}^m$$ . The second aim is to establish an above type theorem with truncation level 1 for differentiably nondegenerate meromorphic mappings.

Interpolation between Brezis–Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities

This paper is devoted to investigate an interpolation inequality between the Brezis–Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and only if it is isometric to a Minkowski space. Our arguments explore fine properties of Bessel functions, comparison principles, and anisotropic symmetrization on Minkowski spaces. As an application, we characterize the existence of nonzero solutions for a quasilinear PDE involving the Finsler–Laplace operator and a Hardy-type singularity on Minkowski spaces where the sharp BPV inequality plays a crucial role. The results are also new in the Riemannian/Euclidean setting.

Holomorphic Vanishing Theorems on Finsler Holomorphic Vector Bundles and Complex Finsler Manifolds

AbstractIn this paper, we investigate the holomorphic sections of holomorphic Finsler bundles over both compact and non-compact complete complex manifolds. We also inquire into the holomorphic vector fields on compact and non-compact complete complex Finsler manifolds. We get vanishing theorems in each case according to different certain curvature conditions. This work can be considered as generalizations of the classical results on Kähler manifolds and hermitian bundles.

On the Uniqueness of Ricci Flow

In this note, we study the problem of the uniqueness of solutions to the Ricci flow on complete noncompact manifolds. We consider the class of solutions with curvature bounded above by C / t when $$t>0$$ and prove a uniqueness result when initial curvature is of polynomial growth and Ricci curvature of the flow is relatively small.

Vanishing theorems for complete Riemannian manifolds with nonnegative scalar curvature

In this paper, we show several vanishing theorems for harmonic p-forms on compact manifolds, and $$L^q$$ harmonic p-forms on complete noncompact manifolds with nonnegative scalar curvature, under various pointwise or integral curvature conditions. These conditions involve the positive Yamabe invariant, the weighted Poincare inequality and the pure curvature tensor.