共查询到20条相似文献,搜索用时 15 毫秒
1.
F. Loose 《Journal of Geometric Analysis》2000,10(2):323-337
Let (X, ω) be a symplectic manifold and ι: M ? X an isotropic embedding, ι*ω = 0. The isotropie embedding theorem gives a local normal form of X in a neighborhood of M, in particular a natural potential α of ω, ?dα = ω. Now, given certain geometrical structures on M and on the symplectic normal bundle of M, in particular inducing a natural energy momentum function H in a neighborhood of M, we construct a natural complex structure J in a neighborhood of M satisfying certain initial conditions associated to the given initial data along M and satisfying the equation (in J): dc H = α. This generalizes a theorem of Guillemin-Stenzel and Lempert-Szöke in the Lagrangean case. 相似文献
2.
Frederick Wilhelm 《Journal of Geometric Analysis》2001,11(3):519-560
In this article we show that there is an exotic sphere with positive sectional curvature almost everywhere.
In 1974 Gromoll and Meyer found a metric of nonnegative sectional on an exotic 7-sphere. They showed that the metric has positive
curvature at a point and asserted, without proof, that the metric has positive sectional curvature almost everywhere [4].
We will show here that this assertion is wrong. In fact, the Gromoll-Meyer sphere has zero curvatures on an open set of points.
Never the less, its metric can be perturbed to one that has positive curvature almost everywhere. 相似文献
3.
Let Ω, ⊂R
n and n ≥ 4 be even. We show that if a sequence {uj} in W1,n/2(Ω;R
n) is almost conformal in the sense that dist (∇uj,R
+SO(n)) converges strongly to 0 in Ln/2 and if uj converges weakly to u in W1,n/2, then u is conformal and ∇uj → ∇u strongly in L
loc
q
for all 1 < -q < n/2. It is known that this conclusion fails if n/2 is replaced by any smaller exponent p. We also prove
the existence of a quasiconvex function f(A) that satisfies 0 ≤ f(A) ≤ C (1 + |A|n/2) and vanishes exactly onR
+ SO(n). The proof of these results involves the Iwaniec-Martin characterization of conformal maps, the weak continuity and
biting convergence of Jacobians, and the weak-L1 estimates for Hodge decompositions. 相似文献
4.
R. Monneau 《Journal of Geometric Analysis》2003,13(2):359-389
We study the obstacle problem in two dimensions. On the one hand we improve a result of L.A. Caffarelli and N.M. Rivière:
we state that every connected component of the interior of the coincidence set has at most N
0
singular points, where N
0
is only dependent on some geometric constants. Moreover, if the component is small enough, then this component has at most
two singular points. On the other hand, we prove in a simple case a conjecture of D.G. Schaeffer on the generic regularity
of the free boundary: for a family of obstacle problems in two dimensions continuously indexed by a parameter λ, the free
boundary of the solution uλ is analytic for almost every λ. Finally we present a new monotonicity formula for singular points.
Dedicated to Henri Berestycki and Alexis Bonnet. 相似文献
5.
Zhongwei Shen 《Journal of Geometric Analysis》2006,16(4):721-734
Using Maz ’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz
and convex domains in ℝn. Forn ≥ 8, combinedwitharesultin[18], these estimates lead to the solvability of the Lp Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the
estimates allow us to show that the Lp Dirichlet problem is uniquely solvable for any 2 − ε < p < ∞ and n ≥ 4. 相似文献
6.
Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and
2
2
(M) be the Sobolev space consisting of functions in L2(M) whose derivatives up to the order two are also in L2(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H
2
2
(M),
where 2#=2n/(n−4) is critical, and
is the usual norm on the Sobolev space H
1
2
(M) consisting of functions in L2(M) whose derivatives of order one are also in L2(M). The sharp constant A in this inequality is K
0
2
where K0, an explicit constant depending only on n, is the sharp constant for the Euclidean Sobolev inequality
. We prove in this article that for any compact Riemannian manifold, A=K
0
2
is attained in the above inequality. 相似文献
7.
Georg Sebastian Weiss 《Journal of Geometric Analysis》1999,9(2):317-326
Regularity of the free boundary ?{u > 0} of a non-negative minimum u of the functional $\upsilon \mapsto \int\limits_\Omega {\left( {\left| {\nabla \upsilon } \right|^2 + Q^2 \chi _{\left\{ {\upsilon > 0} \right\}} } \right)} $ , where Ω is an open set in ?n and Q is a strictly positive Hölder-continuous function, is still an open problem for n ≥ 3. By means of a new monotonicity formula we prove that the existence of singularities is equivalent to the existence of an absolute minimum u* such that the graph of u* is a cone with vertex at 0, the free boundary ?{u* > 0} has one and only one singularity, and the set {u* > 0} minimizes the perimeter among all its subsets. This leads to the following partial regularity: there is a maximal dimension k* ≥ 3 such that for n < k* the free boundary ?{u > 0} is locally in Ω a C1,α-surface, for n = k* the singular set Σ:= ?{u > 0} ? ?red{u > 0} consists at most of in Ω isolated points, and for n > k* the Hausdorff dimension of the singular set Σ is less than n - k*. 相似文献
8.
Abdelhamid Meziani 《Journal of Geometric Analysis》1999,9(2):301-315
This paper deals with the integrability problem for structures on the sphere S2 which are elliptic on the upper and lower hemispheres, and which are given by a simple fold on the equator. Criteria for
standardness are given. Existence and factorization of first integrals are studied. 相似文献
9.
Nahum Zobin 《Journal of Geometric Analysis》1999,9(3):491-511
Consider the Sobolev space W
∞
k
(Ω) of functions with bounded kth derivatives defined in a planar domain. We study the problem of extendability of functions
from W
∞
k
(Ω) to the whole ℝ2 with preservation of class, i.e., surjectivity of the restriction operator W
∞
k
(ℝ2) → W
∞
k
(Ω). 相似文献
10.
In this paper a unique continuation result is proved for differential inequality of second order. 相似文献
11.
Hidetaka Hamada 《Journal of Geometric Analysis》1998,8(3):441-446
The purpose of this paper is to prove that every proper holomorphic self-mapping of a Reinhardt domain Ω in
C
n
which is a generalization of a complex ellipsoid is biholomorphic. The main novelty of our result is that Ω is a domain in
C
n
such that it is allowed to have a boundary point at which the Levi determinant has infinite order of vanishing. 相似文献
12.
We prove smoothness of strictly Levi convex solutions to the Levi equation in several complex variables. This equation is
fully non linear and naturally arises in the study of real hypersurfaces in ℂn+1, for n ≥ 2. For a particular choice of the right-hand side, our equation has the meaning of total Levi curvature of a real
hypersurface ℂn+1 and it is the analogous of the equation with prescribed Gauss curvature for the complex structure. However, it is degenerate
elliptic also if restricted to strictly Levi convex functions. This basic failure does not allow us to use elliptic techniques
such in the classical real and complex Monge-Ampère equations. By taking into account the natural geometry of the problem
we prove that first order intrinsic derivatives of strictly Levi convex solutions satisfy a good equation. The smoothness
of solutions is then achieved by mean of a bootstrap argument in tangent directions to the hypersurface. 相似文献
13.
Baouendi M. S. Mir Nordine Rothschild Linda Preiss 《Journal of Geometric Analysis》2002,12(4):543-580
Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂ
N
are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map
is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic
submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs
of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that
if two real-algebraic hypersurfaces in ℂ
N
are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general
context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds. 相似文献
14.
15.
In this article we analyze viscosity solutions of the one phase Hele-Shaw problem in the plane and the corresponding free
boundaries near a singularity. We find, up to order of magnitude, the speed at which the free boundary moves starting from
a wedge, cusp, or finger-type singularity. Maximum principle-type arguments play a key role in the analysis. 相似文献
16.
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
17.
Christine M. Guenther 《Journal of Geometric Analysis》2002,12(3):425-436
In this article we prove the existence of a fundamental solution for the linear parabolic operator L(u) = (Δ − ∂/∂t − h)u,
on a compact n-dimensional manifold M with a time-parameterized family of smooth Riemannian metrics g(t). Δ is the time-dependent
Laplacian based on g(t), and h(x, t) is smooth. Uniqueness, positivity, the adjoint property, and the semigroup property hold.
We further derive a Harnack inequality for positive solutions of L(u) = 0 on (M, g(t) on a time interval depending on curvature
bounds and the dimension of M, and in the special case of Ricci flow, use it to find lower bounds on the fundamental solution
of the heat operator in terms of geometric data and an explicit Euclidean type heat kernel. 相似文献
18.
Marilyn Daily 《Journal of Geometric Analysis》2007,17(1):75-85
An area minimizing double bubble in ℝn is given by two (not necessarily connected) regions which have two prescribed n-dimensional volumes whose combined boundary
has least (n−1)-dimensional area. The double bubble theorem states that such an area minimizer is necessarily given by a standard
double bubble, composed of three spherical caps. This has now been proven for n = 2, 3,4, but is, for general volumes, unknown
for n ≥ 5. Here, for arbitrary n, we prove a conjectured lower bound on the mean curvature of a standard double bubble. This
provides an alternative line of reasoning for part of the proof of the double bubble theorem in ℝ3, as well as some new component bounds in ℝn. 相似文献
19.
Luca Brandolini Alex Iosevich Giancarlo Travaglini 《Journal of Fourier Analysis and Applications》2001,7(4):359-372
Let Γ be a smooth compact convex planar curve with arc length dm and let dσ=ψ dm where ψ is a cutoff function. For Θ∈SO (2)
set σΘ(E) = σ(ΘE) for any measurable planar set E. Then, for suitable functions f in ℝ2, the inequality.
represents an average over rotations, of the Stein-Tomas restriction phenomenon. We obtain best possible indices for the
above inequality when Γ is any convex curve and under various geometric assumptions. 相似文献
20.
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