Classification of homogeneous holomorphic two-spheres in complex Grassmann manifolds

In this paper we completely classify the linearly full homogeneous holomorphic two-spheres in the complex Grassmann manifolds $G(2,N)$ and $G(3,N)$. We also obtain the Gauss equation for the holomorphic immersions from a Riemann surface into $G(k,N)$. By using which, we give explicit expressions of the Gaussian curvature and the square of the length of the second fundamental form of these homogeneous holomorphic two-spheres in $G(2,N)$ and $G(3,N)$.     

Expanding solitons to the Hermitian curvature flow on complex Lie groups

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on complex Lie groups starting from expanding solitons on their nilradicals.     

Translating solitons in Riemannian products

In this paper we study solitons invariant with respect to the flow generated by a complete parallel vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product $(\mathbb{R}\times P,\mathrm{d}{t}^2+g_0)$ and the parallel field is $X={?}_t$. Similarly to what happens in the Euclidean setting, we call them translating solitons. We see that a translating soliton in $\mathbb{R}\times P$ can be seen as a minimal submanifold for a weighted volume functional. Moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in $\mathbb{R}\times P$. When $g_0$ is rotationally invariant and its sectional curvature is non-positive, we are able to characterize all the rotationally invariant translating solitons. Furthermore, we use these families of new examples as barriers to deduce several non-existence results.     

Metrics with Positive Scalar Curvature at Infinity and Localization Algebra

In this paper, the authors give a new proof of Block and Weinberger’s Bochner vanishing theorem built on direct computations in the K-theory of the localization algebra.     

Erratum to: “Integral curvature bounds and bounded diameter with Bakry–Emery Ricci tensor” [Differ. Geom. Appl. 66 (2019) 42–51]

We found that we need to revise some proofs in the paper “Integral curvature bounds and bounded diameter with Bakry–Emery Ricci tensor”. Hence, we correct them in the present paper.     

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: “When is the Laplace–Beltrami operator , , invertible?” We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103–120). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let be an open and closed subset of the boundary of M. We say that has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance from a point to is bounded uniformly in x (and hence, in particular, intersects all connected components of M). For manifolds with finite width, we prove a Poincaré inequality for functions vanishing on , thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincaré inequality then leads, as in the classical case to results on the spectrum of Δ with domain given by mixed boundary conditions, in particular, Δ is invertible for manifolds with finite width. The bounded geometry assumption then allows us to prove the well‐posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces , .     

基于均匀荷载面曲率的简支裂纹梁损伤识别

为了采用模态参数对结构裂纹进行定位与定量,基于集中柔度模型,采用无质量的扭转弹簧模拟裂纹,建立简支裂纹梁的振动微分方程。针对现有柔度曲率指标仅能判断裂纹的大致范围,基于线性插值理论,建立裂纹位置与相邻测点均匀荷载面曲率差的关系,提出裂纹进一步定位公式,实现裂纹位置的精确定位。针对现有大多数损伤识别方法无法实现裂纹的损伤定量,基于位移曲率与结构刚度和弯矩的关系,理论推导了均匀荷载面曲率的结构刚度损伤程度识别方法,基于弹簧串联原理和线刚度思想,首次提出串联等效线刚度模型,建立裂纹深度与均匀荷载面曲率的关系,实现裂纹深度的定量。通过简支裂纹梁数值算例,考虑多裂纹的损伤情况,验证了新方法对裂纹定位与定量的有效性。     

A note on Chern-Yamabe problem

We propose a flow to study the Chern-Yamabe problem and discuss the long time existence of the flow. In the balanced case we show that the Chern-Yamabe problem is the Euler-Lagrange equation of some functional. The monotonicity of the functional along the flow is derived. We also show that the functional is not bounded from below.     

On mean curvature flow with forcing

Abstract

This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. With the class of forcing which bounds the volume of the evolving set away from zero and infinity, we show that a strong version of star-shapedness is preserved over time. More precisely, it is shown that the flow preserves the ρ-reflection property, which corresponds to a quantitative Lipschitz property of the set with respect to the nearest ball. Based on this property we show that the problem is well-posed and its solutions starting with ρ-reflection property become instantly smooth. Lastly, for a model problem, we will discuss the flow’s exponential convergence to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the approach developed by Feldman-Kim to combine viscosity solutions approach and variational method. The main challenge lies in the lack of comparison principle, which accompanies forcing terms that penalize small volume.