共查询到20条相似文献,搜索用时 15 毫秒
1.
We consider the minimization problem for an average distance functional in the plane, among all compact connected sets of
prescribed length. For a minimizing set, the blow-up sequence in the neighborhood of any point is investigated. We show existence
of the blow up limits and we characterize them, using the results to get some partial regularity statements. 相似文献
2.
Frank Morgan 《Journal of Geometric Analysis》2005,15(2):321-341
In (the surface of) a convex polytope P
3 inR
4,an areaminimizing surface avoids the vertices of P and crosses the edges orthogonally.
In a smooth Riemannian manifold M with a group of isometries G, an areaminimizing G-invariant oriented hypersurface is smooth
(except for a very small singular set in high dimensions). Already in 3D, area-minimizing G-invariant unoriented surfaces
can have certain singularities, such as three orthogonal sheets meeting at a point. We also treat other categories of surfaces
such as rectifiable currents modulo v and soap films. 相似文献
3.
This paper proves (i) every “geometrically knotted” non-closed curve bounds a soap-film, (ii) any non-closed curve bounding
a soap-film must have total curvature greater than 2π, and (iii) for every k > 2π, there is a geometrically knotted non-closed
curve with total curvature k. 相似文献
4.
Frank Morgan 《Journal of Geometric Analysis》2007,17(1):97-106
In (the surface of) a convex polytope Pn in ℝn+1, for small prescribed volume, geodesic balls about some vertex minimize perimeter. 相似文献
5.
Evans M. Harrell 《Journal of Geometric Analysis》2002,12(1):81-88
The Blaschke-Lebesgue Theorem states that among all planar convex domains of given constant width B the Reuleaux triangle
has minimal area. It is the purpose of this article to give a direct proof of this theorem by analyzing the underlying variational
problem. The advantages of the proof are that it shows uniqueness (modulo rigid deformations such as rotation and translation)
and leads analytically to the shape of the area-minimizing domain. Most previous proofs have relied on foreknowledge of the
minimizing domain. Key parts of the analysis extend to the higher-dimensional situation, where the convex body of given constant
width and minimal volume is unknown. 相似文献
6.
A soap film is actually a thin solid fluid bounded by two surfaces of opposite orientation. It is natural to model the film
using one polyhedron for each side. Two problems are to get the polyhedra for both sides to be in the same place without canceling
each other out and to model triple junctions without introducing extra boundary components. We use chainlet geometry to create
dipole cells and mass cells which accomplish these goals and model faithfully all observable soap films and bubbles. We introduce
a new norm on chains of these cells and prove lower semicontinuity of area. A geometric version of Carton’s magic formula
provides the necessary boundary coherence. 相似文献
7.
A measurable set Q ⊂
R
n
is a wavelet set for an expansive matrix A if F
−1
(ΧQ) is an A-dilation wavelet. Dai, Larson, and Speegle [7] discovered the existence of wavelet sets in
R
n
associated with any real n ×n expansive matrix. In this work, we construct a class of compact wavelet sets which do not contain the origin and which are,
up to a certain linear transformation, finite unions of integer translates of an integral selfaffine tile associated with
the matrix B = A
t. Some of these wavelet sets may have good potential for applications because of their tractable geometric shapes. 相似文献
8.
We prove existence and almost everywhere regularity of an area minimizing soap film with a bound on energy spanning a given
Jordan curve in Euclidean space R
3.The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of
surfaces over which area is minimized includes images of disks, integral currents, nonorientable surfaces and soap films as
observed by Plateau with a bound on energy. Our area minimizing solution is shown to be a smooth surface away from its branched
set which is a union of Lipschitz Jordan curves of finite total length. 相似文献
9.
Thierry De Pauw 《Journal of Geometric Analysis》2002,12(1):29-61
A concentrated (ξ, m) almost monotone measure inR
n
is a Radon measure Φ satisfying the two following conditions: (1) Θ
m
(Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R
n
the ratioexp [ξ(r)]r−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim
r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R
n
and a λ-Lipschitz function f from x+W into x+W‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ. 相似文献
10.
Given a compact, oriented Riemannian manifold M, without boundary, and a codimension-one homology class in H* (M, Z) (or, respectively, in H* (M, Zp) with p an odd prime), we consider the problem of finding a cycle of least area in the given class: this is known as the
homological Plateau’s problem.
We propose an elliptic regularization of this problem, by constructing suitable fiber bundles ξ (resp. ζ) on M, and one-parameter
families of functionals defined on the regular sections of ξ, (resp. ζ), depending on a small parameter ε.
As ε → 0, the minimizers of these functionals are shown to converge to some limiting section, whose discontinuity set is exactly
the minimal cycle desired. 相似文献
11.
V. Yaskin 《Journal of Geometric Analysis》2006,16(4):735-745
The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in ℝn with smaller volume of all k-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer
to this question is negative if k>3. The problem is still open for k = 2, 3. In this article we formulate and completely solve
the lower dimensional Busemann-Petty problem in the hyperbolic space ℍn. 相似文献
12.
Manuel Ritoré 《Journal of Geometric Analysis》2001,11(3):509-517
We show that in a complete plane with nonnegative curvature there is a perimeter minimizing set of any given area. This set
is a disc whose boundary is a closed embedded curve with constant geodesic curvature. 相似文献
13.
We give the first existence and regularity results on the cheapest way to enclose and separate planar regions of prescribed
areas, where cost is given by a general norm ϕ, thus generalizing the Wulff shape for enclosing a single region. As an example,
we classify the cheapest way to enclose and separate two planar regions of prescribed areas for the ℓ1 norm (“Manhattan metric”) into three distinct types, according to the relative size of the prescribed areas. 相似文献
14.
Let
be the unit disk of the complex plane. A conformai map of
into itself is called hyperbolically convex if the non-Euclidean segment between any two points of
also belongs to
. In this paper we prove several inequalities that are analogous to inequalities about (Euclidean) convex univalent functions.
We show that if ƒ (0) = 0, then Re zf′/f > 1/2. This inequality is the key for the results of this paper. In particular we
deduce a three-variable inequality corresponding to that of Ruscheweyh and Sheil-Small. The sharp bound for the Schwarzian
derivative remains open. 相似文献
15.
Keomkyo Seo 《Archiv der Mathematik》2008,90(2):173-180
Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a nonpositive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C. If ∂Σ ∼ ∂C is radially connected from a point , then we prove a sharp relative isoperimetric inequality
where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove the relative isoperimetric inequalities for minimal submanifolds outside a closed convex set in a higher-dimensional
Riemannian manifold.
Received: 3 February 2007 相似文献
16.
In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ
n
which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal
vector fields in ℍ
n
.We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean
curvature (CMC) hypersurface. Our definition coincides with previous ones.
Our main result describes which are the CMC hypersurfaces of revolution in ℍ
n
.The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential
equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart
in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean
space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ
n
. 相似文献
17.
Kenneth A. Brakke 《Journal of Geometric Analysis》1995,5(4):445-514
A new mathematical model of soap films is proposed, called the “covering space model.” The two sides of a film are modeled
as currents on different sheets of a covering space branching along the film boundary. Hence a film may be seen as the minimal
cut separating one sheet of the covering space from the others. The film is thus the oriented boundary of one sheet, which
represents the exterior of the film. As oriented boundaries, films may be calibrated with differential forms on the covering
space, a version of the min-cut, max-flow duality of network theory. This model applies to unoriented films, films with singularities,
films touching only part of a knotted curve, films that deformation retract to their boundaries, and other examples that have
proved troublesome for previous soap film models.
Communicated by Frederick Almgren 相似文献
18.
A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric
inequalities of plane convex geometry: The Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential
geometry. 相似文献
19.
The double coset space AΛ (n, ℂ) / U (n − 1, 1) is studied, where A consists of the diagonal matrices in GL (n, ℂ). This space
naturally arises in the harmonic analysis on the hermitian symmetric space GL (n, ℂ) / U (n − 1, 1). It is shown here that
these double cosets also represent a class of basic invariants related to complex hyperbolic geometry. An algebraic parametrization
for the double cosets is given and it is shown how this may be used to conveniently compute the geometric invariants. 相似文献
20.
A colorful theorem on transversal lines to plane convex sets 总被引:1,自引:0,他引:1
We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the
plane has the property that any three differently colored members have a transversal line that meet the sets consistently
with the numbering, then there exists a color such that all the convex sets of that color have a transversal line.
All authors are partially supported by CONACYT research grant 5040017. 相似文献