将闭Riemann子流形保角变形成极小子流形

在本文中，通过外围空间的适当保角变形，我们证明了，每个Riemann子流形可以被认作一个极小子流形，我们还研究了这样得到的子流形的稳定性，定理2和3推广了Schoen和S．TYan[2]的结论．     

Volume Growth,Number of Ends,and the Topology of a Complete Submanifold

Given a complete isometric immersion φ:P ^m ?N ⁿ in an ambient Riemannian manifold N ⁿ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space  \(M^{n}_{w}\) , we determine a set of conditions on the extrinsic curvatures of P that guarantee that the immersion is proper and that P has finite topology in line with the results reported in Bessa et al. (Commun. Anal. Geom. 15(4):725–732, 2007) and Bessa and Costa (Glasg. Math. J. 51:669–680, 2009). When the ambient manifold is a radially symmetric space, an inequality is shown between the (extrinsic) volume growth of a complete and minimal submanifold and its number of ends, which generalizes the classical inequality stated in Anderson (Preprint IHES, 1984) for complete and minimal submanifolds in ? ⁿ . As a corollary we obtain the corresponding inequality between the (extrinsic) volume growth and the number of ends of a complete and minimal submanifold in hyperbolic space, together with Bernstein-type results for such submanifolds in Euclidean and hyperbolic spaces, in the manner of the work Kasue and Sugahara (Osaka J. Math. 24:679–704, 1987).     

Lower Bounds for The Volume of the Image of a Ball

Ukrainian Mathematical Journal - We consider ring Q-homeomorphisms with respect to the p-modulus in the space ?n for p>n. A lower bound for the volume of the image of a ball under...     

Mean Curvature and Volume of Extrinsic Spheres in Hypersurfaces of Real Space Forms

Given a hypersurface P^n-1 in a real space form of constantcurvature b, , we have obtained a lower bound for the norm of the mean curvature normal vector field of extrinsicspheres in P^n-1 in terms of the mean curvature of the geodesic spheres in , with the same radius, and the meancurvature of P^n-1, characterizing too the equality.     

The Jump of the Laplacian on a Submanifold

Assume that a submanifold M ? ?ⁿ of an arbitrary codimension k ? {1, …, n} is closed in some open set O→?ⁿ. With a given function u ? C²(O\M) we may associate its trivial extension u: O→? such that u|_O\M=u and u|m ≡ 0. The jump of the Laplacian of the function u on the submanifold M is defined by the distribution Δu — Δu. By applying some general version of the Fubini theorem to the nonlinear projection onto M we obtain the formula for the jump of the Laplacian (Theorem 2.2).     

Residues of Holomorphic Foliations Relative to a General Submanifold

Let F be a holomorphic foliation (possibly with singularities)on a non-singular manifold M, and let V be a complex analyticsubset of M. Usual residue theorems along V in the theory ofcomplex foliations require that V be tangent to the foliation(that is, a union of leaves and singular points of V and F);this is the case for instance for the blow-up of a non-dicriticalisolated singularity. In this paper, residue theorems are introducedalong subvarieties that are not necessarily tangent to the foliation,including the blow-up of the dicritical situation. 2000 MathematicsSubject Classification 53C12, 57R20, 55N15.     

Geometry of the Normal Bundle of a Submanifold*

 We study the geometric behavior of the normal bundle T ^⊥ M of a submanifold M of a Riemannian manifold . We compute explicitely the second fundamental form of T ^⊥ M and look at the relation between the minimality of T ^⊥ M and M. Finally we show that the Maslov forms with respect to a suitable connection of the pair (T ^⊥ M, are null. Received March 14, 2001; in revised form February 11, 2002     

Brownian Motion and the Distance to a Submanifold

This is a study of the distance between a Brownian motion and a submanifold of a complete Riemannian manifold. It contains a variety of results, including an inequality for the Laplacian of the distance function derived from a Jacobian comparison theorem, a characterization of local time on a hypersurface which includes a formula for the mean local time, an exit time estimate for tubular neighbourhoods and a concentration inequality. The concentration inequality is derived using moment estimates to obtain an exponential bound, which holds under fairly general assumptions and which is sufficiently sharp to imply a comparison theorem. We provide numerous examples throughout. Further applications will feature in a subsequent article, where we see how the main results and methods presented here can be applied to certain study objects which appear naturally in the theory of submanifold bridge processes.     

欧氏空间中子流形的Pinching现象

本文研究了欧氏空间中紧致子流形的Ｐｉｎｃｈｉｎｇ现象,得到了一些公式,并证明了一些几何量的Ｐｉｎｃｈｉｎｇ定理     

Brownian Motion and Riemannian Geometry in the Neighbourhood of a Submanifold

Let M be a complete connected smooth Riemannian manifold of dimension n and P a q-dimensional smoothly embedded smooth submanifold of M. M₀ will denote a tubular neighbourhood of P in M. Let L = \(\frac{1}{2}\Delta\) + b + c be a differential operator on M, where Δ is the Laplacian on smooth functions, b a smooth vector field on M and c a smooth potential term. Let p\(_{t}^{\mathrm{M}_{0}}(-,-)\) be the Dirichlet heat kernel of M₀, and p\(_{t}^{\mathrm{M}}(-,-)\) the heat kernel of M. We will show in this article that for a smooth function f:M→R with compact support in M₀, the integral \(\int_{\mathrm{P}}\)f(y)p\(_{t}^{\mathrm{M}_{0}}\)(x,y)π(dy) generalizes the usual Dirichlet heat kernel and has an asymptotic expansion of the form:

$ \int_{\mathrm{P}}f(y)p_{t}^{\mathrm{M}_{0}}(x,y)\pi({\rm dy}) = q_{t} (x,P)\left[ \mathrm{f}(\gamma(\mathrm{t}))+\sum\limits_{\alpha=1}^{N}\mathrm{b}_{\alpha}\mathrm{(x,P)t}^{\alpha}+ \mathrm{o}(t^{N})\right] , $

where π is the Riemannian measure on P and q _t (x,P) is defined in Eq. 2.7. The asymptotic expansion is then extended to \(\int_{\mathrm{P}}\)f(y)p\(_{t}^{\mathrm{M}}\)(x,y)π(dy). The above expansion generalizes the usual Minakshisundaram–Pleijel heat kernel expansion and a computation of the leading expansion coefficients suggests that it is also a generalization of the heat content expansion. The expansion coefficients are local geometric invariants given by simple integrals of the derivatives of the metric tensor and the volume change factor θ _P . The leading coefficients are then computed in terms of the Riemannian geometry in the neighbourhood of the submanifold P at the centre of Fermi coordinates y₀?∈?P.

     

The Stokes Problem in a Ball and in the Exterior of a Ball

We analyze the formula for the solution to the Stokes problem in a ball and in the exterior of a ball which was derived in the previous paper by the author published in 2002. The main result is pointwise estimates for the kernels of potentials in this formula. Bibliography: 4 titles.     

具有平行平均曲率向量的子流形的二次表示

本文证明了具有平行平均曲率向量的子流形的二次表示不可能是零２型的，以及若具有平行平均曲率向量的ａ－子流形的二次表示是２－型的，则它的数量曲率必为常数．     

一类流形的刚性定理

本文通过具有良好性质的子流形的存在性,证明了一类流形的一个刚性定理,并得到形如Ｂｏｎｎｅｔ－Ｍｙｅｒｓ定理的推论．我们还指出,在主要定理中全测地子流形的条件一般不能减弱为极小子流形．     

P-Sasakian流形的外蕴球面(英)

本文讨论了P－Sasakian流形的外蕴球面，并给出了P－Sasakian流形的外蕴球面的一个分类定理．     

极小子流形刚性的若干结果

利用欧氏空间子流形上的Bochner公式,结合极小子流形上存在的L2-Sobolev不等式,将Ni Lei的具有上界"total scalar curvature"的极小超曲面的刚性定理的结果推广到极小子流形的情形,并得到了关于极小子流形的一个曲率估计.     

黎曼流形中紧子流形的拼挤定理

本利用几何不等式和曲率估计的方法,证明了黎曼流形N^n＋p,上的具有平行平均曲率的紧子流形M^n上的一个拼挤定理。若N上的截曲率KN满足- 1≤ KN≤δ≤0,且‖S- nH2‖n/2, ‖ S-nH^2‖n/n-s满足一些不等式,则δ= - 1。     

常曲率空间中的正曲率子流形

本文研究了常曲率空间子流表余维数减少的问题,说明了在一定条件下,余维数可以减少到１。     

The Logarithmic Sobolev Inequality for a Submanifold in Manifolds with Nonnegative Sectional Curvature

The authors prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifolds with nonnegative sectional curvature of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature. This extends a recent result of Brendle with Euclidean setting.     

An Expression for the Riemann Tensor of a Submanifold Given by a System of Equations in a Riemannian Space

We derive an expression for the Riemann tensor of a submanifold given implicitly by a system of independent equations in a Riemannian space. In particular, we prove a formula for the internal curvature of a two-surface in a three-dimensional Riemannian space. Some applications of the formula are given.