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1.
Ye Li 《Annals of Global Analysis and Geometry》2012,41(4):407-421
This note is a continuation of the author’s paper (Li, Adv. Math. 223(6):1924–1957, 2010). We prove that if the metric g of a compact 4-manifold has bounded Ricci curvature and its curvature has no local concentration everywhere, then it can
be smoothed to a metric with bounded sectional curvature. Here we don’t assume the bound for local Sobolev constant of g and hence this smoothing result can be applied to the collapsing case. 相似文献
2.
Theodoros Vlachos 《manuscripta mathematica》2008,126(2):201-230
We deal with minimal surfaces in a sphere and investigate certain invariants of geometric significance, the Hopf differentials,
which are defined in terms of the complex structure and the higher fundamental forms. We discuss the holomorphicity of Hopf
differentials and provide a geometric interpretation for it in terms of the higher curvature ellipses. This motivates the
study of a class of minimal surfaces, which we call exceptional. We show that exceptional minimal surfaces are related to
Lawson’s conjecture regarding the Ricci condition. Indeed, we prove that, under certain conditions, compact minimal surfaces
in spheres which satisfy the Ricci condition are exceptional. Thus, under these conditions, the proof of Lawson’s conjecture
is reduced to its confirmation for exceptional minimal surfaces. In fact, we provide an affirmative answer to Lawson’s conjecture
for exceptional minimal surfaces in odd dimensional spheres or in S
4m
. 相似文献
3.
In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery
Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula
for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions.
In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact
Riemannian manifolds. 相似文献
4.
Daniel Maerten 《Annals of Global Analysis and Geometry》2007,32(4):391-414
We prove a Penrose-like inequality for the mass of a large class of constant mean curvature (CMC) asymptotically flat n-dimensional spin manifolds which satisfy the dominant energy condition and have a future converging, or past converging compact
and connected boundary of non-positive mean curvature and of positive Yamabe invariant. We prove that for every n ≥ 3 the mass is bounded from below by an expression involving the norm of the linear momentum, the volume of the boundary,
dimensionless geometric constants and some normalized Sobolev ratio. 相似文献
5.
I. G. Korepanov 《Theoretical and Mathematical Physics》2009,158(1):82-95
Geometric torsions are torsions of acyclic complexes of vector spaces consisting of differentials of geometric quantities
assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a three-dimensional
manifold with a triangulated boundary. These invariants can be naturally combined into a vector, and a change of the boundary
triangulation corresponds to a linear transformation of this vector. Moreover, when two manifolds are glued at their common
boundary, these vectors undergo scalar multiplication, i.e., they satisfy Atiyah’s axioms of a topological quantum field theory.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 98–114, January, 2009. 相似文献
6.
I. G. Korepanov 《Theoretical and Mathematical Physics》2009,158(3):344-354
We generalize the construction of invariants of three-dimensional manifolds with a triangulated boundary that we previously
proposed for the case where the boundary consists of not more than one connected component to any number of components. These
invariants are based on the torsion of acyclic complexes of geometric origin. An adequate tool for studying such invariants
turns out to be Berezin’s calculus of anticommuting variables; in particular, they are used to formulate our main theorem,
concerning the composition of invariants under a gluing of manifolds. We show that the theory satisfies a natural modification
of Atiyah’s axioms for anticommuting variables.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 405–418, March, 2009. 相似文献
7.
Joseph H.G. Fu 《manuscripta mathematica》1998,97(2):175-187
A rectifiable current of dimension n−1 in the sphere bundle Sℝn≃ℝn×S
n
−1 for euclidean space is Legendrian if it annihilates the contact 1-form α (i.e. T(α∧φ)=0 for all forms φ of degree n−2). Such a current may be naturally associated to any convex set or to any singular real analytic variety, and induces the
curvature measures of such a set. We prove that the projection to ℝn of a carrier of a general such T is C
2-rectifiable in the sense of Anzellotti–Serapioni. We deduce that the boundary of a set with positive reach, as well as its
singular skeleta, are C
2-rectifiable.
In case ∂T= 0 we prove also that the curvature measures associated to T satisfy the analogues of the classical variational formulas for curvature integrals. It follows that such formulas are valid
for the curvature measures of subsets of space forms.
Received: 3 December 1997/ Revised version: 25 May 1998 相似文献
8.
We prove Khinchin’s Theorems for Gelfand pairs (G, K) satisfying a condition (*): (a)G is connected; (b)G is almost connected and Ad (G/M) is almost algebraic for some compact normal subgroupM; (c)G admits a compact open normal subgroup; (d) (G,K) is symmetric andG is 2-root compact; (e)G is a Zariski-connectedp-adic algebraic group; (f) compact extension of unipotent algebraic groups; (g) compact extension of connected nilpotent groups.
In fact, condition (*) turns out to be necessary and sufficient forK-biinvariant measures on aforementioned Gelfand pairs to be Hungarian. We also prove that Cramér’s theorem does not hold for
a class of Gaussians on compact Gelfand pairs.
This author was supported by the European Commission (TMR 1998–2001 Network Harmonic Analysis). 相似文献
9.
Chengbo Yue 《Israel Journal of Mathematics》1997,97(1):113-123
We answer a question of Gromov ([G2]) in the codimension 1 case: ifF is a codimension 1 foliation of a compact manifoldM with leaves of negative curvature, thenπ
1(M) has exponential growth. We also prove a result analogous to Zimmer’s ([Z2]): ifF is a codimension 1 foliation on a compact manifold with leaves of nonpositive curvature, and ifπ
1(M) has subexponential growth, then almost every leaf is flat. We give a foliated version of the Hopf theorem on surfaces without
conjugate points.
Partially supported by NSF Grant #DMS 9403870. 相似文献
10.
For a conic linear system of the form Ax ∈K, K a convex cone, several condition measures have been extensively studied in the last dozen years. Among these, Renegar’s condition
number is arguably the most prominent for its relation to data perturbation, error bounds, problem geometry, and computational complexity
of algorithms. Nonetheless, is a representation-dependent measure which is usually difficult to interpret and may lead to overly conservative bounds
of computational complexity and/or geometric quantities associated with the set of feasible solutions. Herein we show that
Renegar’s condition number is bounded from above and below by certain purely geometric quantities associated with A and K; furthermore our bounds highlight the role of the singular values of A and their relationship with the condition number. Moreover, by using the notion of conic curvature, we show how Renegar’s
condition number can be used to provide both lower and upper bounds on the width of the set of feasible solutions. This complements
the literature where only lower bounds have heretofore been developed. 相似文献
11.
Rafael López 《Monatshefte für Mathematik》1999,76(1):155-169
We study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary (=H-surfaces). We prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples. When the surface is embedded, conditions to be umbilical are given. Finally, we characterize umbilical surfaces bounded by a circle among all H-discs with small area. 相似文献
12.
We consider discrete cocompact isometric actions where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces — complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov — as Hadamard
spaces) and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We
identify invariants (“geometric data”) of the action which determine, and are determined by, the equivariant homeomorphism type of the action of G on the ideal boundary of X. Moreover, if are two actions with the same geometric data and is a G-equivariant quasi-isometry, then for every geodesic ray there is a geodesic ray (unique up to equivalence) so that . This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136].
Submitted: May 2001. 相似文献
13.
Thomas Lorenz 《Set-Valued Analysis》2008,16(1):1-50
The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order
geometric evolutions, however, the topological boundary need not be continuous in the sense of Painlevé–Kuratowski. So this
paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional
analysis beyond the traditional border of vector spaces: Distribution-like solutions are introduced in a set just supplied
with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler
approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological
vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of ℝ
N
evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity
assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions
for the uniqueness of these solutions.
相似文献
14.
S. V. Kharitonova 《Journal of Mathematical Sciences》2011,177(5):742-747
In this paper, we study almost C(λ)-manifolds. We obtain necessary and sufficient conditions for an almost contact metric manifold to be an almost C(λ)-manifold. We prove that contact analogs of A. Gray’s second and third curvature identities on almost C(λ)-manifolds hold, while a contact analog of A. Gray’s first identity holds if and only if the manifold is cosymplectic.
It is proved that a conformally flat, almost C(λ)-manifold is a manifold of constant curvature λ. 相似文献
15.
16.
Christian Bär 《Annals of Global Analysis and Geometry》2009,36(1):67-79
We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds.
This includes Friedrich’s estimate for manifolds with positive scalar curvature as well as the author’s estimate on surfaces.
相似文献
17.
We prove that there exist (n − 1)-dimensional compact embedded rotational hypersurfaces with constant scalar curvature (n − 1)(n − 2)S (S > 1) of S
n
other than product of spheres for 4 ≤ n ≤ 6. As a result, we prove that Leite’s Assertion was incorrect.The project is supported by the grant No. 10531090 of NSFC. 相似文献
18.
19.
We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem
1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate.
As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also
show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive
time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ
n
.In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s
entropy in the case of Riemann surfaces. 相似文献
20.
Rafael López 《Monatshefte für Mathematik》1999,127(2):155-169
We study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary (=H-surfaces). We prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples. When the surface is embedded, conditions to be umbilical are given. Finally, we characterize
umbilical surfaces bounded by a circle among all H-discs with small area.
Received 27 March 1997; in final form 11 June 1998 相似文献