共查询到20条相似文献,搜索用时 46 毫秒
1.
Martin Reiris 《Annales Henri Poincare》2010,10(8):1559-1604
Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume
V(k)=(\frac-k3)3Volg(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by
Vinf=(\frac-16Y(S))\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g
i
, K
i
)} satisfying: (i) k
i
= −3, (ii) Viˉ Vinf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q
0((g
i
, K
i
)) ≤ Λ, where Q
0 is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state
is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples
for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and
cosmologically normalized flow
([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)2g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,¥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E1)\tilde]=E1(([(g)\tilde],[(K)\tilde])) £ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E1=Q0+Q1{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ Vinf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}. 相似文献
2.
In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called
BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal
rotational hypersurfaces generated by plane curves rotating around the axis in the direction of
[(b)\tilde]\sharp{\tilde{\beta}^{\sharp}} in Minkowski (α, β)-space
(\mathbbVn+1,[(Fb)\tilde]){(\mathbb{V}^{n+1},\tilde{F_b})} , where
\mathbbVn+1{\mathbb{V}^{n+1}} is an (n+1)-dimensional real vector space, [(Fb)\tilde]=[(a)\tilde]f([(b)\tilde]/[(a)\tilde]), [(a)\tilde]{\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}} is the Euclidean metric, [(b)\tilde]{\tilde{\beta}} is a one form of constant length
b:=||[(b)\tilde]||[(a)\tilde], [(b)\tilde]\sharp{b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}} is the dual vector of [(b)\tilde]{\tilde{\beta}} with respect to [(a)\tilde]{\tilde{\alpha}} . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated
around the axis in the direction of
[(b)\tilde]\sharp{\tilde{\beta}^{\sharp}} in Minkowski Randers 3-space
(\mathbbV3,[(a)\tilde]+[(b)\tilde]){(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})} . 相似文献
3.
Manuel del Pino Michal Kowalczyk Juncheng Wei Jun Yang 《Geometric And Functional Analysis》2010,20(4):918-957
Let (M,[(g)\tilde]){(\mathcal {M},\tilde{g})} be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation
e2 D[(g)\tilde] u + (1 - u2 )u = 0 in M,\varepsilon ^2 \Delta _{\tilde g} u \, + \, (1 - u^2 )u\, =\, 0 \quad {\rm{in}} \, \mathcal {M}, 相似文献
4.
Dani Szpruch 《The Ramanujan Journal》2011,26(1):45-53
Let
\mathbbF\mathbb{F} be a p-adic field, let χ be a character of
\mathbbF*\mathbb{F}^{*}, let ψ be a character of
\mathbbF\mathbb{F} and let gy-1\gamma_{\psi}^{-1} be the normalized Weil factor associated with a character of second degree. We prove here that one can define a meromorphic
function [(g)\tilde](c,s,y)\widetilde{\gamma}(\chi ,s,\psi) via a similar functional equation to the one used for the definition of the Tate γ-factor replacing the role of the Fourier transform with an integration against y·gy-1\psi\cdot\gamma_{\psi}^{-1}. It turns out that γ and [(g)\tilde]\widetilde{\gamma} have similar integral representations. Furthermore, [(g)\tilde]\widetilde{\gamma} has a relation to Shahidi‘s metaplectic local coefficient which is similar to the relation γ has with (the non-metalpectic) Shahidi‘s local coefficient. Up to an exponential factor, [(g)\tilde](c,s,y)\widetilde{\gamma}(\chi,s,\psi) is equal to the ratio
\fracg(c2,2s,y)g(c,s+\frac12,y)\frac{\gamma(\chi^{2},2s,\psi)}{\gamma(\chi,s+\frac{1}{2},\psi)}. 相似文献
5.
Jakob Jonsson 《Annals of Combinatorics》2010,14(4):487-505
Let 1 ≤ m ≤ n. We prove various results about the chessboard complex M
m,n
, which is the simplicial complex of matchings in the complete bipartite graph K
m,n
. First, we demonstrate that there is nonvanishing 3-torsion in
[(H)\tilde]d(\sf Mm,n; \mathbb Z){{\tilde{H}_d({\sf M}_{m,n}; {\mathbb Z})}} whenever
\fracm+n-43 £ d £ m-4{{\frac{m+n-4}{3}\leq d \leq m-4}} and whenever 6 ≤ m < n and d = m − 3. Combining this result with theorems due to Friedman and Hanlon and to Shareshian and Wachs, we characterize all triples
(m, n, d ) satisfying
[(H)\tilde]d (\sf Mm,n; \mathbb Z) 1 0{{\tilde{H}_d \left({\sf M}_{m,n}; {\mathbb Z}\right) \neq 0}}. Second, for each k ≥ 0, we show that there is a polynomial f
k
(a, b) of degree 3k such that the dimension of
[(H)\tilde]k+a+2b-2 (\sf Mk+a+3b-1,k+2a+3b-1; \mathbb Z3){{\tilde{H}_{k+a+2b-2}}\,\left({{\sf M}_{k+a+3b-1,k+2a+3b-1}}; \mathbb Z_{3}\right)}, viewed as a vector space over
\mathbbZ3{\mathbb{Z}_3}, is at most f
k
(a, b) for all a ≥ 0 and b ≥ k + 2. Third, we give a computer-free proof that
[(H)\tilde]2 (\sf M5,5; \mathbb Z) @ \mathbb Z3{{\tilde{H}_2 ({\sf M}_{5,5}; \mathbb {Z})\cong \mathbb Z_{3}}}. Several proofs are based on a new long exact sequence relating the homology of a certain subcomplex of M
m,n
to the homology of M
m-2,n-1 and M
m-2,n-3. 相似文献
6.
We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}
7.
Christopher Hammond 《Mathematische Zeitschrift》2010,266(2):285-288
Let Ω be a domain in ${\mathbb{C}^{2}}
8.
We establish two new lower bounds for the halfspace range searching problem: Given a set of n points in ℝ
d
, where each point is associated with a weight from a commutative semigroup, compute the semigroup sum of the weights of the
points lying within any query halfspace. Letting m denote the space requirements, we prove a lower bound for general semigroups of
[\varOmega\tilde](n1-1/(d+1)/m1/(d+1))\widetilde{\varOmega}(n^{1-1/(d+1)}/m^{1/(d+1)}) and for integral semigroups of
[\varOmega\tilde](n/m1/d)\widetilde{\varOmega}(n/m^{1/d}). 相似文献
9.
Leonid G. Fel 《Functional Analysis and Other Mathematics》2011,3(2):179-192
We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87−4k) and S(9,3+9k,85−9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r12,r1r2+r12k,r3-r12k)\mathsf{S}(r_{1}^{2},r_{1}r_{2}+r_{1}^{2}k,r_{3}-r_{1}^{2}k), k∈ℤ, r
1,r
2,r
3∈ℤ+, r
1≥2 and gcd(r
1,r
2)=gcd(r
1,r
3)=1, and calculate their universal Frobenius number Φ(r
1,r
2,r
3) for the wide range of k providing semigroups be symmetric. We show that this type of semigroups admit also nonsymmetric representatives. We describe
the reduction of the minimal generating sets of these semigroups up to {r12,r3-r12k}\{r_{1}^{2},r_{3}-r_{1}^{2}k\} for sporadic values of k and find these values by solving the quadratic Diophantine equation. 相似文献
10.
Tapani Matala-aho 《Constructive Approximation》2011,33(3):289-312
We shall present short proofs for type II (simultaneous) Hermite–Padé approximations of the generalized hypergeometric and
q-hypergeometric series
|