共查询到20条相似文献,搜索用时 15 毫秒
1.
The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces
always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results
hold for the dual space .
(Received 27 August 1999; in revised form 18 November 1999) 相似文献
2.
We provide a series of examples of Riemannian manifoldsequipped with a minimal unit vector field. 相似文献
3.
We provide a new characterization of invariant minimal unit vector fields on Lie groups and use it to construct some new examples. In particular, we determine all these vector fields on three-dimensional Lie groups. 相似文献
4.
We consider the energy (or the total bending) of unit vector fields oncompact Riemannian manifolds for which the set of its singularitiesconsists of a finite number of isolated points and a finite number ofpairwise disjoint closed submanifolds. We determine lower bounds for theenergy of such vector fields on general compact Riemannian manifolds andin particular on compact rank one symmetric spaces. For this last classof spaces, we compute explicit expressions for the total bending whenthe unit vector field is the gradient field of the distance function toa point or to special totally geodesic submanifolds (i.e., for radialunit vector fields around this point or these submanifolds). 相似文献
5.
It is well known that no non-trivial Killing vector field existson a compact Riemannian manifold of negative Ricci curvature;analogously, no non-trivial harmonic one-form exists on a compactmanifold of positive Ricci curvature. One can consider the following,more general, problem. By reducing the assumption on the Riccicurvature to one on the scalar curvature, such vanishing theoremscannot hold in general. This raises the question: "What informationcan we obtain from the existence of non-trivial Killing vectorfields (or, respectively, harmonic one-forms)?" This paper givesanswers to this problem; the results obtained are optimal. 2000Mathematics Subject Classification 53C20 (primary), 53C24 (secondary). 相似文献
6.
Bo?ek(1980) has introduced a class of solvable Lie groups Gnwith arbitrary odd dimension to construct irreducible generalized symmetric Riemannian space such that the identity component of its full isometry group is solvable. In this article, the authors provide the set of all left-invariant minimal unit vector fields on the solvable Lie group Gn,and give the relationships between the minimal unit vector fields and the geodesic vector fields, the strongly normal unit vectors respectively. 相似文献
7.
Potential Analysis - In this article, we continue our investigations of the boundary behavior of harmonic mappings. We first discuss the classical problem on the growth of radial length and obtain... 相似文献
8.
The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Gr?tzsch and Johannes C.C.?Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bj?rling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope. 相似文献
9.
All the two-dimensional space charge equipotential surfaceswhich are congruent to the equipotential surfaces of equivalentharmonic problems are determined. The known list of solutionsof the high-pressure space charge equation is thereby increased.A voltage-current characteristic derived from the new solutionsis compared with experimental data for unipolar DC transmissionlines. 相似文献
10.
11.
Sebastian Walcher 《Results in Mathematics》1997,31(1-2):161-169
We discuss one case where the integration of a sum of vector fields is reducible to the integration of the summands. Applications include the construction of a class of additive group actions on affine space and a proof that these are stably tame, and also the explicit solution of a class of differential equations from mathematical biology. 相似文献
12.
The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results hold for the dual space . 相似文献
13.
The main result of the paper is the following generalization of Forelli’s theorem (Math. Scand. 41:358–364, 1977): Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with eigenvalues whose ratios are positive reals. Then any function φ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p.
We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary.
K.T. Kim and G. Schmalz were supported by the Scientific visits to Korea program of the AAS and KOSEF. E. Poletsky was supported
by NSF Grant DMS-0500880. G. Schmalz gratefully acknowledges support and hospitality of the Max-Planck-Institut für Mathematik
Bonn. 相似文献
14.
Murolo C.; Du Plessis A. A.; Trotman D. J. A. 《Journal London Mathematical Society》2005,71(2):516-530
For a stratification with suitable regularity, in particularfor any Whitney stratification and, via regular embedding, forany abstract stratified set, time-dependent vector fields areused to prove an extension theorem for diffeomorphisms nearthe identity defined on strata of a given dimension. Then itis shown that after isotopy a stratified map h : Z can bemade transverse to a fixed stratified map g : y . 相似文献
15.
Some relations between the causal character of projective vector fields and curvature on a Lorentzian manifold M are studied. As a consequence, obstructions to the existence of such vector fields are found. Affine, homothetic and Killing vector fields are considered specifically. 相似文献
16.
We provide techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and their consequences) into more effective control. In the present paper, we focus on proving regularity theorems for minimizing harmonic maps and minimal currents. There are several aspects to our improvements of known estimates. First, we replace known estimates on the Hausdorff dimension of singular sets by estimates on the volumes of their r‐tubular neighborhoods. Second, we give improved regularity control with respect to the number of derivatives bounded and/or on the norm in which the derivatives are bounded. As an example of the former, our results for minimizing harmonic maps $f: M^n \rightarrow N^m $ between Riemannian manifolds include a priori bounds in $ W^{1,p} \cap W^{2,{p}/{2}} $ for all p < 3. These are the first such bounds involving second derivatives in general dimensions. Finally, the quantity we control actually provides much stronger information than that which follows from a bound on the Lp norm of derivatives. Namely, we obtain Lp bounds for the reciprocal of the regularity scale $r_f(x):= \max\{r: \sup_{B_r(x)}r|\nabla f|+r^2|\nabla^2 f|\leq 1\}$ . Applications to minimal hypersufaces include a priori Lp bounds for the second fundamental form A for all p < 7. Previously known bounds were for $ p < 4+ \sqrt{{8}/{n}} $ in the smooth immersed stable case. Again, the full theorem is much stronger and yields Lp bounds for the reciprocal of the corresponding regularity scale $r_{|A|}(x):= \max\{r: \sup_{B_r(x)}r|A|\leq 1\}$ . In outline, our discussion follows that of an earlier paper in which we proved analogous estimates in the context of noncollapsed Riemannian manifolds with a lower bound on Ricci curvature. These were applied to Einstein manifolds. A key role in each of these arguments is played by the relevant quantitative differentiation theorem. © 2013 Wiley Periodicals, Inc. 相似文献
17.
For any given vector field X defined on some open set M 2, we characterize the prolongations X
n
*
of X to the nth jet space M
(n), n1, such that a complete system of invariants for X
n
*
can be obtained by derivation of lower-order invariants. This leads to characterizations of C
-symmetries and to new procedures for reducing the order of an ordinary differential equation. 相似文献
18.
J. H. Wang G. López V. Martín-Márquez C. Li 《Journal of Optimization Theory and Applications》2010,146(3):691-708
The relationship between monotonicity and accretivity on Riemannian manifolds is studied in this paper and both concepts are
proved to be equivalent in Hadamard manifolds. As a consequence an iterative method is obtained for approximating singularities
of Lipschitz continuous, strongly monotone mappings. We also establish the equivalence between the strong convexity of functions
and the strong monotonicity of its subdifferentials on Riemannian manifolds. These results are then applied to solve the minimization
of convex functions on Riemannian manifolds. 相似文献
19.
Hubert Gollek 《Results in Mathematics》2007,50(1-2):53-79
20.
Olav K. Richter 《Journal of Number Theory》2002,95(1):101-121
We investigate theta functions attached to quadratic forms over a number field K. We establish a functional equation by regarding the theta functions as specializations of symplectic theta functions. By applying a differential operator to the functional equation, we show how theta functions with harmonic coefficients over K behave under modular transformations. 相似文献