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This paper deals with an initial-boundary value problem for the chemotaxis-(Navier–)Stokes system $$\begin{aligned} \left\{ \begin{array}{lcll} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot (n\nabla c), \qquad &{}\quad x\in \Omega , \ t>0, \\ c_t + u\cdot \nabla c &{}=&{}\Delta c - nc, \qquad &{}\quad x\in \Omega , \ t>0, \\ u_t + \kappa (u\cdot \nabla )u &{}=&{} \Delta u - \nabla P + n\nabla \phi , \qquad &{}\quad x\in \Omega , \ t>0, \\ \nabla \cdot u =0, &{} &{} \qquad &{}\quad x\in \Omega , t>0, \end{array} \right. \end{aligned}$$ in a bounded convex domain $$\Omega \subset \mathbb {R}^2$$ with smooth boundary, with $$\kappa \in \mathbb {R}$$ and a given smooth potential $$\phi :\Omega \rightarrow \mathbb {R}$$ . It is known that for each $$\kappa \in \mathbb {R}$$ and all sufficiently smooth initial data this problem possesses a unique global classical solution $$(n^{(\kappa )},c^{(\kappa )},u^{(\kappa }))$$ . The present work asserts that these solutions stabilize to $$(n^{(0)},c^{(0)},u^{(0)})$$ uniformly with respect to the time variable. More precisely, it is shown that there exist $$\mu >0$$ and $$C>0$$ such that whenever $$\kappa \in (-1,1)$$ , $$\begin{aligned}&\Big \Vert n^{(\kappa )}(\cdot ,t)-n^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} + \Big \Vert c^{(\kappa )}(\cdot ,t)-c^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} \\&\quad +\, \Big \Vert u^{(\kappa )}(\cdot ,t)-u^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} \le C |\kappa | e^{-\mu t} \end{aligned}$$ for all $$t>0$$ . This result thereby provides an example for a rigorous quantification of stability properties in the Stokes limit process, as frequently considered in the literature on chemotaxis-fluid systems in application contexts involving low Reynolds numbers.

In this paper, we introduce a new type rearrangement inequality based on the Steiner rearrangement. To show orbital stability of standing wave solutions with respect to a nonlinear Schrodinger equation, \(H^1\)-precompactness of minimizing sequence of \(L^2\)-constraint minimizing problem is very important. Usually, by using the concentration compactness principle and some scaling argument, \(H^1\)-precompactness is obtained. To prove \(H^1\)-precompactness without scaling arguments, we introduce the rearrangement. The Steiner rearrangement is defined as a map from \(H^1\) to \(H^1\), whereas our rearrangement \((\cdot \star \cdot )\) is defined as a map from \(H^1 \times H^1\) to \(H^1\). By using the rearrangement, we show a strict inequality \(\Vert \nabla (u \star v) \Vert _{L^2}^2 < \Vert \nabla u\Vert _{L^2}^2 + \Vert \nabla v\Vert _{L^2}^2\) under simple assumptions.

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2n-1$ in $\mathbb{C}^{N}$. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For $n\ge 3$ and $N=n+1$, Yau found a necessary and sufficient condition for the interior regularity of the Harvey-Lawson solution to the complex Plateau problem by means of Kohn--Rossi cohomology groups on $X$ in 1981. For $n=2$ and $N\ge n+1$, the first and third authors introduced a new CR invariant $g^{(1,1)}(X)$ of $X$. The vanishing of this invariant will give the interior regularity of the Harvey-Lawson solution up to normalization. For $n\ge 3$ and $N>n+1$, the problem still remains open. In this paper, we generalize the invariant $g^{(1,1)}(X)$ to higher dimension as $g^{(\Lambda^n 1)}(X)$ and show that if $g^{(\Lambda^n 1)}(X)=0$, then the interior has at most finite number of rational singularities. In particular, if $X$ is Calabi--Yau of real dimension $5$, then the vanishing of this invariant is equivalent to give the interior regularity up to normalization., Comment: arXiv admin note: substantial text overlap with arXiv:1203.1380

We consider foliations of the whole three dimensional hyperbolic space $${\mathbb {H}}^{3}$$ by oriented geodesics. Let $${\mathcal {L}}$$ be the space of all the oriented geodesics of $${\mathbb {H}}^{3}$$ , which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics of signature $$\left( 2,2\right) $$ . We characterize, in terms of these geometries of $${\mathcal {L}}$$ , the subsets $${\mathcal {M}}$$ in $${\mathcal {L}}$$ that determine foliations of $${\mathbb {H}}^{3}$$ . We describe in a similar way some distinguished types of geodesic foliations of $${\mathbb {H}}^{3}$$ , regarding to which extent they are in some sense trivial in some directions: On the one hand, foliations whose leaves do not lie in a totally geodesic surface, not even at the infinitesimal level. On the other hand, those for which the forward and backward Gauss maps $$\varphi ^{\pm }:{\mathcal {M}}\rightarrow {\mathbb {H}} ^{3}\left( \infty \right) $$ are local diffeomorphisms. Besides, we prove that for this kind of foliations, $$\varphi ^{\pm }$$ are global diffeomorphisms onto their images. The subject of this article is within the framework of foliations by congruent submanifolds, and follows the spirit of the paper by Gluck and Warner where they understand the infinite dimensional manifold of all the great circle foliations of the three sphere.

Recently, we proved that every finite dimensional Alexandrov space is strongly locally Lipschitz contractible. In the present paper, we consider the set $$\mathcal M$$ of all isometry classes of Alexandrov spaces of curvature $$\ge -1$$ and of fixed dimension having upper diameter bound and lower volume bound, and prove that there exists a constant $$N$$ depending on the parameters determining $$\mathcal M$$ such that every space in $$\mathcal M$$ can be covered by at most $$N$$ strongly Lipschitz contractible balls. Also, we prove that there exists a constant $$N^\prime $$ depending on $$\mathcal M$$ such that every space in $$\mathcal M$$ can be covered by at most $$N^\prime $$ strongly Lipschitz contractible and convex regions.

The aim of this paper is to prove a Mahler measure formula of a four-variable Laurent polynomial whose zero locus defines a Calabi–Yau threefold. We show that its Mahler measure is a rational linear combination of a special \(L\)-value of the normalized newform in \(S_4(\Gamma _0(8))\) and a Riemann zeta value. This is equivalent to a new formula for a \(_6F_5\)-hypergeometric series evaluated at 1.

Let $$F$$ be a non-archimedean local field, $$\pi $$ an admissible irreducible $$\mathbf{GL }_{2}(F)$$ -representation with complex coefficients. For a quadratic extension $$L/F$$ and an $$L^{\times }$$ -character $$\chi $$ a classical result of Tunnell and Saito establish a precise connection between the dimension of the Hom-space $$\hbox {Hom}_{L^{\times }}\big (\pi \vert _{L^{\times }},\chi \big )$$ and the normalized local factor of the pair $$(\pi ,\chi )$$ . The study of analogous Hom-spaces for complex valued representations has recently been generalized to $$\mathbf{GL}_n$$ in Aizenbud et al. (Ann Math 172:1407–1434, 2010) and their connections with local factors have been established by work of Moeglin and Waldspurger (La conjecture locale de Gross–Prasad pour les groupes speciaux orthogonaux: le cas general, preprint. Available at http://www.math.jussieu.fr/moeglin/gp.pdf , 2009). In this paper we approach the analogous problem in the context of the $$p$$ -modular Langlands correspondence for $$\mathbf{GL }_{2}({\mathbf{Q}}_p)$$ . We describe the restriction to Cartan subgroups of an irreducible $$p$$ -modular representation $$\pi $$ of $$\mathbf{GL }_{2}({\mathbf{Q}}_p)$$ and deduce generalized multiplicity results on the dimension of the Ext-spaces $$\mathrm{Ext}_{{{{\fancyscript{O}}}}_L^{\times }}^{i}\big (\pi \vert _{{{{\fancyscript{O}}}}_L^{\times }},\chi \big )$$ where $${{{\fancyscript{O}}}}_L^{\times }$$ is the ring of integers of a quadratic extension of $${\mathbf{Q}}_p$$ and $$\chi $$ a smooth character of $${{{\fancyscript{O}}}}_L^{\times }$$ .

Bourgain posed the problem of calculating $$\begin{aligned} \Sigma = \sup _{n \ge 1} ~\sup _{k_1 < \cdots < k_n} \frac{1}{\sqrt{n}} \left\| \sum _{j=1}^n e^{2 \pi i k_j \theta } \right\| _{L^1([0,1])}. \end{aligned}$$ It is clear that \(\Sigma \le 1\); beyond that, determining whether \(\Sigma < 1\) or \(\Sigma =1\) would have some interesting implications, for example concerning the problem whether all rank one transformations have singular maximal spectral type. In the present paper we prove \(\Sigma \ge \sqrt{\pi }/2 \approx 0.886\), by this means improving a result of Karatsuba. For the proof we use a quantitative two-dimensional version of the central limit theorem for lacunary trigonometric series, which in its original form is due to Salem and Zygmund.

Let \(M\) be a smooth closed spin (resp. oriented and totally non-spin) manifold of dimension \(n\ge 5\) with fundamental group \(\pi \). It is stated, e.g. by Rosenberg and Stolz (Surv Surg Theory 2, pp. 353–370, 2001), that \(M\) admits a metric of positive scalar curvature (pscm) if its orientation class in \(ko_n(B\pi )\) (resp. \(H_n(B\pi ;\mathbb Z )\)) lies in the subgroup consisting of elements which contain pscm representatives. This is \(2\)-locally verified loc. cit. and by Stolz (Topology 33, pp. 159–180, 1994). After inverting \(2\) it was announced that a proof would be carried out by Jung (uncompleted Ph.D. thesis), but this work has never appeared in print. The purpose of our paper is to present a self-contained proof of the statement with \(2\) inverted.

We consider the equation \(\text{ div}\,\mathbb{Y }=f\), with \(f\) a zero average function on the torus \(\mathbb{T }^d\). In their seminal paper, Bourgain and Brezis [J Am Math Soc 16(2):393–426, 2003 (electronic)] proved the existence of a solution \(\mathbb{Y }\in W^{1,d}\cap L^\infty \) for a datum \(f\in L^d\). We extend their result to the critical Sobolev spaces \(W^{s,p}\) with \((s+1)p=d\) and \(p\ge 2\). More generally, we prove a similar result in the scale of Triebel–Lizorkin spaces. We also consider the equation \(\text{ div} \,\mathbb{Y }=f\) in a bounded domain \(\varOmega \) subject to zero Dirichlet boundary condition.

Using a sifting-shadowing combination, we prove in this paper that an arbitrary C1-class local diffeomorphism f of a closed manifold M n is uniformly expanding on the closure $${\mathrm{Cl}_{M^n}(\mathrm{Per}(f))}$$ of its periodic point set Per(f), if it is nonuniformly expanding on Per(f).

If M is a submanifold of a space form, the nullity distribution \({\mathcal {N}}\) of its second fundamental form is (when defined) the common kernel of its shape operators. In this paper we will give a local description of any submanifold of the Euclidean space by means of its nullity distribution. We will also show the following global result: if M is a complete, irreducible submanifold of the Euclidean space or the sphere then its complementary distribution \({\mathcal {N}^{\bot}}\) is completely non integrable. This means that any two points in M can be joined by a curve everywhere perpendicular to \({\mathcal {N}}\) . We will finally show that this statement is false for a submanifold of the hyperbolic space.

Let $${x: M^{m} \rightarrow \mathbb{S}^{m+1}}$$ be an m-dimensional umbilic-free hypersurface in an (m + 1)-dimensional unit sphere $${\mathbb{S}^{m+1}}$$ , with standard metric I = dx · dx. Let II be the second fundamental form of isometric immersion x. Define the positive function $${\rho=\sqrt{\frac{m}{m-1}}\|II-\frac{1}{m}tr(II)I\|}$$ . Then positive definite (0,2) tensor $${\mathbf{g}=\rho^{2}I}$$ is invariant under conformal transformations of $${\mathbb{S}^{m+1}}$$ and is called Mobius metric. The curvature induced by the metric g is called Mobius curvature. The purpose of this paper is to classify the hypersurfaces with constant Mobius curvature.

The paper establishes a solution to the Monge problem in ℝ for a possibly asymmetric norm cost function and absolutely continuous initial measures, under the assumption that the unit ball is strictly convex-but not necessarily differentiable nor uniformly convex. The proof follows the strategy initially proposed by Sudakov in 1976, found to be incomplete in 2000; the missing step is fixed in the above case adapting a disintegration technique introduced for a variational problem. By strict convexity, mass moves along rays, and we also investigate the divergence of the vector field of rays. © 2010 Springer-Verlag.

Our concern in this paper lies with two aspects of weighted exponential spaces connected with their role of target spaces for critical imbeddings of Sobolev spaces. We characterize weights which do not change an exponential space up to equivalence of norms. Specifically, we first prove that $${L_{\exp t^{\alpha}}(\chi_B)=L_{\exp t^{\alpha}}(\rho)}$$ if and only if $${\rho^q \in L_q}$$ with some q > 1. Second, we consider the Sobolev space $${W^{1}_{N}(\varOmega),}$$ where Ω is a bounded domain in $${\mathbb{R}^{N}}$$ with a sufficiently smooth boundary, and its imbedding into a weighted exponential Orlicz space $${L_{\exp t^{p'}}(\varOmega,\rho)}$$ , where ρ is a radial and non-increasing weight function. We show that there exists no effective weighted improvement of the standard target $${L_{\exp t^{N'}}(\varOmega)=L_{\exp t^{N'}}(\varOmega,\chi_{\varOmega})}$$ in the sense that if $${W^{1}_{N}(\varOmega)}$$ is imbedded into $${L_{\exp t^{p'}}(\varOmega,\rho),}$$ then $${L_{\exp t^{p'}}(\varOmega,\rho)}$$ and $${L_{\exp t^{N'}}(\varOmega)}$$ coincide up to equivalence of the norms; that is, we show that there exists no effective improvement of the standard target space. The same holds for critical cases of higher-order Sobolev spaces and even Besov and Lizorkin–Triebel spaces.

Let $${{\mathbb{Q}^4}(c)}$$ be a four-dimensional space form of constant curvature c. In this paper we show that the infimum of the absolute value of the Gauss–Kronecker curvature of a complete minimal hypersurface in $${\mathbb{Q}^4(c), c\leq 0}$$ , whose Ricci curvature is bounded from below, is equal to zero. Further, we study the connected minimal hypersurfaces M 3 of a space form $${{\mathbb{Q}^4}(c)}$$ with constant Gauss–Kronecker curvature K. For the case c ≤ 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurfaces of $${\mathbb{Q}^4(c)}$$ with K constant.

In this paper we study submanifold with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if $K\subset (S^n, g)$ is a totally geodesic submanifold in a Riemannian sphere with positive sectional curvature where $n\ge 5$, then $K$ is homeomorphic to $S^{n-2}$ and the fundamental group of the knot complement $\pi_1(S^n-K)\cong \Bbb Z$., Comment: 9 pages

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface with non-negative Gaussian curvature and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian product metric $=_M-dt^2$. In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain $\Omega\subseteq M$ is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi-Bernstein result for entire maximal graphs in $M^2\times\mathbb{R}_1$., Comment: Final version (October 2009). To appear in Mathematische Zeitschrift. Dedicated to Professor Marcos Dajczer on the occasion of his 60th birthday

Let k be a finite field of characteristic p, l a prime number different from p, \({\psi : k \to \overline{\bf Q}_l^\ast}\) a nontrivial additive character, and \({\chi : {k^\ast}^n \to \overline{\bf Q}_l^\ast}\) a character on \({{k^\ast}^n}\). Then ψ defines an Artin-Schreier sheaf \({\mathcal{L}_\psi}\) on the affine line \({{\bf A}_k^1}\), and χ defines a Kummer sheaf \({\mathcal{K}_\chi}\) on the n-dimensional torus \({{\bf T}_k^n}\) . Let \({f \in k[X_{1},X_{1}^{-1},\ldots, X_{n},X_n^{-1}]}\) be a Laurent polynomial. It defines a k-morphism \({f : {\bf T}_k^n \to {\bf A}_k^1}\) . In this paper, we calculate the weights of \({H_c^i({\bf T}_{\bar k}^n, {\mathcal K}_\chi \otimes f^\ast{\mathcal L}_\psi)}\) under some non-degeneracy conditions on f. Our results can be used to estimate sums of the form $$\sum_{x_1,\ldots, x_n\in k^\ast} \chi_1(f_1(x_1,\ldots, x_n))\cdots \chi_m(f_m(x_1,\ldots, x_n))\psi(f(x_1,\ldots, x_n)),$$ where \({\chi_1,\ldots, \chi_m : k^\ast\to {\bf C}^\ast}\) are multiplicative characters, \({\psi:k\to {\bf C}^\ast}\) is a nontrivial additive character, and f1 , . . . , fm, f are Laurent polynomials.

Let $${\Omega\subset\mathbb{C}^n}$$ be a bounded pseudoconvex domain with C k boundary, k ≥ 1. In this paper, we will prove that the Cauchy–Riemann operator $$\overline \partial$$ has a bounded solution operator in the Sobolev space $${W^s_{(p,q)}(\Omega)}$$ for all $${0 \leq s < k-\frac{1}{2}}$$ .

This paper concerns the number Zn of sites visited up to time n by a random walk Sn having zero mean and moving on the d-dimensional square lattice Zd. Asymptotic evaluation of the conditional expectation of Zn given that S0 = 0 and Sn = x is carried out under 2 + δ moment conditions (0 ≤ δ ≤ 2) in the cases d = 2, 3. It gives an explicit form of the leading term and reasonable estimates of the remainder term (depending on δ) valid uniformly in each parabolic region of (x, n). In the case x = 0 the problem has been studied for the simple random walk and its analogue for Brownian motion; the estimates obtained here are finer than or comparable to those found in previous works.

Let V be a smooth projective threefold of general type. Denote by K3, a rational number, the self-intersection of the canonical sheaf of any minimal model of V. One defines K3 as a canonical volume of V. The paper is devoted to proving the sharp lower bound \({K^{3} \ge \frac{1}{420}}\) which can be reached by an example: \({X_{46} \subseteq \mathbb{P}(4,5,6,7,23)}\).

Let $C$ be a genus 2 curve and $\su$ the moduli space of semi-stable rank 2 vector bundles on $C$ with trivial determinant. In \cite{bol:wed} we described the parameter space of non stable extension classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf $\omega$ of $C$ with $\omega_C^{-1}$. In this paper we study the classifying rational map $\phi: \pr Ext^1(\omega,\omega^{-1})\cong \pr^4 \dashrightarrow \su\cong \pr^3$ that sends an extension class on the corresponding rank two vector bundle. Moreover we prove that, if we blow up $\pr^4$ along a certain cubic surface $S$ and $\su$ at the point $p$ corresponding to the bundle $\OO \oplus \OO$, then the induced morphism $\tilde{\phi}: Bl_S \ra Bl_p\su$ defines a conic bundle that degenerates on the blow up (at $p$) of the Kummer surface naturally contained in $\su$. Furthermore we construct the $\pr^2$-bundle that contains the conic bundle and we discuss the stability and deformations of one of its components., Comment: 29 pages

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a regular domain G of R^3. We prove that the second fundamental form of the boundary surface of G is nonnegatively defined at every point of the limit set of such immersions., 8 pages, 3 figures. To appear in Math. Z

In this paper we investigate vector-valued parabolic initial boundary value problems \({(\mathcal A(t,x,D)}\) , \({\mathcal B_j(t,x,D))}\) subject to general boundary conditions in domains G in \({\mathbb R^n}\) with compact C 2m -boundary. The top-order coefficients of \({\mathcal A}\) are assumed to be continuous. We characterize optimal L p -L q -regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on \({\mathcal A}\) and the Lopatinskii–Shapiro condition on \({(\mathcal A, \mathcal B_1,\dots, \mathcal B_m)}\) are necessary for these L p -L q -estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.

For a smooth curve C it is known that a very ample line bundle \({\mathcal{L}}\) on C is normally generated if Cliff(\({\mathcal{L}}\)) < Cliff(C) and there exist extremal line bundles \({\mathcal{L}}\) (:non-normally generated very ample line bundle with Cliff(\({\mathcal{L}}\)) = Cliff(C)) with \({h^{1}(\mathcal{L}) \le 1}\) . However it has been unknown whether there exists an extremal line bundle \({\mathcal{L}}\) with \({h^{1}(\mathcal{L}) \ge 2}\) . In this paper, we prove that for any positive integers (g, c) with g = 2c + 5 and \({c \equiv 0}\) (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle \({\mathcal{L}}\) with \({h^{1}(\mathcal{L}) = 2}\) . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle \({\mathcal{L}}\) with \({h^{1}(\mathcal{L}) = 2}\) . More generally, if C has a line bundle \({\mathcal{M}}\) computing the Clifford index c of C with \({(3c/2) + 3 < {\deg} \mathcal{M} \leq g-1}\) , then C has such an extremal line bundle \({\mathcal{L}}\).

In this paper, we will examine a strong form of Oka’s lemma which provides sufficient conditions for compact and subelliptic estimates for the $${{\overline\partial}}$$ -Neumann operator on Lipschitz domains. On smooth domains, the condition for subellipticity is equivalent to D’Angelo finite type and the condition for compactness is equivalent to Catlin’s condition (P). As an application, we will prove regularity for the $${{\overline\partial}}$$ -Neumann operator in the Sobolev space W s , $${0\leq s < \frac{1}{2}}$$ , on C 2 domains.

Triebel’s conjecture on the entropy numbers for the limiting embedding case Bp,p1/p↪ Eν, Eν being suitable Orlicz spaces, is almost proved for p≥2 – almost in the sense that a log -factor occurs in the upper estimate. Furthermore, it is shown that the “breaking point”, where the radii of the ɛ-balls in the covering of the Bp,q1/p-unit ball stop to be constant, essentially depends on the second parameter q. The methods used in the paper also give improvements of the results till now known in the case p

The purpose of this paper is to study singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Under some assumptions on a domain Ω in ℂ n+1 , the Bergman kernel B(z) of Ω takes the form near a boundary point p: $${{ B(z)= \frac{{\Phi(w,\rho)}}{{\rho^{{2+2/d_F}} (\hbox{{log}}(1/\rho))^{{m_F-1}}}}, }}$$ where (w,ρ) is some polar coordinates on a nontangential cone Λ with apex at p and ρ means the distance from the boundary. Here Φ admits some asymptotic expansion with respect to the variables ρ 1/m and log(1/ρ) as ρ→0 on Λ. The values of d F >0, m F ∈ℤ+ and m∈ are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of Φ as ρ→0 on Λ is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szego kernel.

This paper is concerned with a Cauchy problem $ \mbox{(P)} \;\;\; \left \{ \begin{array}{ll} u _t = \Delta u + u^p & \quad \mbox{ in }{\bf R}^N \times (0, \infty), u (x,0) = \lambda \varphi(x) & \quad \mbox{ in } {\bf R}^N, \end{array} \right.$ where $ p > p_{\ast} \equiv (N+2)/(N-2), \lambda > 0 $ and $ \varphi$ is a nonnegative radially symmetric function in $ C^1({\bf R}^N) $ with compact support. Denote the solution of (P) by $u_{\lambda} $ . Let $p^{\ast} = \infty $ if $ 3 \leq N \leq 10 $ and $p^{\ast} = 1+6/(N-10) $ if $ N \geq 11 $. We show that if $ p_{\ast} < p < p^{\ast} $, then there is $ \lambda_{\varphi} > 0 $ such that: (i) If $ \lambda < \lambda_{\varphi} $, then $ u_{\lambda} $ exists globally in time in the classical sense and $ u_{\lambda}(t) $ converges to zero locally uniformly in ${\bf R}^N $ as $ t \to \infty $ . (ii) If $ \lambda = \lambda_{\varphi} $ , then $ u_{\lambda} $ blows upincompletely in finite time. (iii) If $ \lambda > \lambda_{\varphi} $ , then $ u_{\lambda} $ blows upcompletely in finite time.

In this paper we compute the generating function for the Euler characteristic of the Deligne-Mumford compactification of the moduli space of smooth n-pointed genus 2 curves. The proof relies on quite elementary methods, such as the enumeration of the graphs involved in a suitable stratification of \(\overline{{\cal M}}_{2,n}\).

Let (X,L) be a polarized manifold with dim X = n. In this paper, we classify (X,L) with n = 3, $h^{0}(L)\geq 5$ , and g(L)=q(X) + 2. Moreover we also classify (X,L) with $n\geq 3$ , g(L)=q(x) + 2, and $\operatorname{Bs}|L|=\emptyset$ .

Let V be a Euclidean Jordan algebra with an associative inner product hji; and let be the corresponding symmetric cone. Then the linear automorphism group H of the symmetric cone acts transitively on it. Let T := V + i be the symmetric tube domain associated to the symmetric cone ; and let G(T) be the biholomorphic automorphism group of T: There exists a conformal compactification M := G(T)=P of V with respect to a parabolic subgroup P: A Lie semigroup that is naturally related to the symmetric cone occurs as the semigroup of compressionsof in the Lie group G(T): = fg 2 G(T) j g() g: It turns out that admits an Ol’shanskii decomposition = H(expC) for an Ad(H)-invariant convex cone C in the Lie algebra g(T) of G(T); a triple decomposition = + H ([8], [9]), and it determines a globally hyperbolic causal order on the homogeneous space G(T)=H defined by aH bH if and only if a 1 b 2 : The family of bilinear forms x(u; v )= hP(x) 1 ujvi defines a Hinvariant Riemannian metric on ; where the map P is the quadratic representation of the Jordan algebra V: In [8], Koufany showed that the elements of are contractions of for the Riemannian metric . The main result of this paper is that the elements of the interior of (called, strict compressions) are strict contractions of for the Riemannian distance induced by

An orientation reversing involution \(\sigma\) of a topological compact genus \(g,\, g>2,\) surface \(\Sigma\) induces an antiholomorphic involution \(\sigma^*: T^g \to T^g\) of the Teichmuller space of genus g Riemann surfaces. Two such involutions \(\sigma^*\) and \(\tau^*\) are conjugate in the mapping class group if and only if the corresponding orientation reversing involutions \(\sigma\) and \(\tau\) of \(\Sigma\) are conjugate in the automorphism group of \(\Sigma\). This is equivalent to saying that the quotient surfaces \(\Sigma/\langle \sigma\rangle\) and \(\Sigma/\langle \tau\rangle\) are homeomorphic. Hence the Teichmuller space \(T^g\) has \(m_g = \lfloor{3g+4\over 2}\rfloor\) distinct antiholomorphic involutions, which are also called real structures of \(T^g\) ([7]). This result is a simple fact that follows from Royden's theorem ([4]) stating that the the mapping class group is the full group of holomorphic automorphisms of the Teichmuller space (\(g>2\)). Let \(\sigma^*: T^g\to T^g\) and \(\tau^*: T^g\to T^g\) be two real structures that are not conjugate in the mapping class group. In this paper we construct a real analytic diffeomorphism \(d: T^g\to T^g\) such that

In [10] Huneke noted a proof due to Hochster of the following theorem of Angeniol and Giusti (see [1]). Let R be a local ring, n>m, and g: R"~R " a homomorphism such that grade(Im(g))=n-m+l. Suppose the cokernel N of g has finite length. Then this length equals the length of R/Ira(g): )~(N)= 2(R/Ira(g)). By I,,(g) we denote the ideal in R generated by the m x m minors of a matrix representing g. It is well-known (see [7]) that n m + 1 is the maximal possible grade if Im(g ) :t = R. Therefore R is a Cohen-Macaulay ring of dimension n m + 1. We further notice that Angeniol and Giusti, in addition, assume R to contain the rational numbers and that the theorem has been known for a long time if m=n ([9], 21.10.17.2). In this paper we shall study the problem of Angeniol and Giusti in a more general context. We start from the "dual" situation, f : Rm~R" being a homomorphism, which we assume to be injective (this is equivalent to grade (Ira(f))> 1). Given a basis _x of R m and an orientation 7 on R", i.e. an

The importance of exhaustion functions in complex analysis arises mainly from the theorems of Andreott i and Grauert [-2] and the applications of Morse theory. The first give vanishing theorems for cohomology groups of coherent sheaves, the second imply topological statements: If X is a compact complex manifold and Y an analytic subset of X, the vanishing of the relative homotopy groups rh(X, Y) for i < d i m X q follows from the q-completeness of X Y , which is by definition the same as the existence of a q-convex exhaustion function. These theorems are called Lefschetz-theorems. The first of this kind was the Lefschetz-theorem on hyperplane sections, which can be proved by using the fact that the complement of a hypersurface in projective space is Stein, because there is a 1-convex exhaustion function. It is natural to ask whether there is an exhaustion function of P , A with certain properties if one has a Lefschetz theorem. If A is a q-codimensional complex submanifold of P, we have H~(P,, A; 7Z,)=0 ([-4]) and even ~i(P,, A ) = 0 ([9]) for i< n 2 q + 1. So we are led to the question if ]P , -A is (2 q 1)-complete, and the proof of this assertion is the aim of this paper.

In this paper we will prove several Sobolev type inequalities for holomorphic functions in strictly pseudoconvex domains. To a large extent, this work was motivated by the following problem. Let D be a strictly pseudoconvex domain and let M be the intersection with D of a submanifold of a neighborhood o f / ) which intersects 0D transversally. Let f be a holomorphic function on M lying in some L p class (e.g. a Hardy or Bergman class). We want to find a holomorphic extension of f to D satisfying an optimal L p estimate on D. The case p --c~ was solved by Henkin [14] who showed that a bounded holomorphic function on M is necessarily the restriction of a bounded holomorphic function on D. For certain weighted Bergman spaces on M, optimal L p estimates for extensions were obtained by Cumenge [9] in the case l < p < o o and by Beatrous [4] in the case 0 < p < c~. More precisely, letting 6(z) denote the distance from z to 0D, the following result was proved. Let f be a holomorphic function on M satisfying