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In this paper, we consider a sequence of orbifold Kahler–Einstein metrics \((\omega _{X_m})_m\) and normalized Kahler–Ricci flows \((\omega _{X_m}(t))_m\) on a quasi-projective manifold \(X=\overline{X} {\setminus } \overline{D}\) for a projective manifold \(\overline{X}\) and a divisor \(\overline{D}\) of \(\overline{X}\) with simple normal crossings such that \(K_{\overline{X}}+\overline{D}\) is ample. For sufficiently large \(m, \omega _{X_m}\) or \(\omega _{X_m}(t)\) is the orbifold Kahler–Einstein metric or orbifold normalized Kahler–Ricci flow on \(X\) equipped with the orbifold structure with respect to the divisor \(1/m\overline{D}\), respectively. The main theorem of this paper is that the limit on the boundary \(\overline{D}\) or the inside \(X\) of the sequence \((\omega _{X_m}(t))_m\) as \(m \rightarrow \infty \) is exactly the complete normalized Kahler–Ricci flow on \(\overline{D}\) or \(X\), respectively. Moreover our method of the proof also leads to a result that on the boundary \(\overline{D}\) the sequence \((\omega _{X_m})_m\) converges to the complete Kahler–Einstein metric on \(\overline{D}\) as \(m \rightarrow \infty \).

In this paper, we prove the local converse theorem for $${\mathrm {Sp}}_{2r}(F)$$ over a p-adic field F. More precisely, given two irreducible supercuspidal representations of $${\mathrm {Sp}}_{2r}(F)$$ with the same central character such that they are generic with the same additive character and they have the same gamma factors when twisted with generic irreducible representations of $${\mathrm {GL}}_n(F)$$ for all $$1\le n\le r$$ , then these two representations must be isomorphic. Our proof is based on the local analysis of the local integrals which define local gamma factors. A key ingredient of the proof is certain partial Bessel function property developed by Cogdell–Shahidi–Tsai recently. The same method can give the local converse theorem for $${\mathrm {U}}(r,r)$$ .

Let $$G=\mathrm{GL}_{2n}$$ over a totally real number field F and $$n\ge 2$$ . Let $$\Pi $$ be a cuspidal automorphic representation of $$G(\mathbb {A})$$ , which is cohomological and a functorial lift from SO $$(2n+1)$$ . The latter condition can be equivalently reformulated that the exterior square L-function of $$\Pi $$ has a pole at $$s=1$$ . In this paper, we prove a rationality result for the residue of the exterior square L-function at $$s=1$$ and also for the holomorphic value of the symmetric square L-function at $$s=1$$ attached to $$\Pi $$ . As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L-functions and a product of Gaus sums of Hecke characters.

Let $$\varOmega $$ be a smooth bounded domain in $$\mathbb {R}^N$$ , $$N\ge 3$$ . We consider the following singularly perturbed nonlinear elliptic problem on $$\varOmega $$ , $$\begin{aligned} \varepsilon ^2\varDelta v-v+f(v)=0,\quad v>0\ \text {on}\ \varOmega ,\qquad \frac{\partial v}{\partial \nu }=0\quad \text {on}\ \partial \varOmega , \end{aligned}$$ where $$\nu $$ is an exterior unit normal vector to $$\partial \varOmega $$ and a nonlinearity f satisfies subcritical growth condition. It has been known that for any $$l_0, l_1 \in \mathbb {N} \cup \{ 0 \}$$ , $$l_0+l_1>0$$ , there exists a solution $$v_\varepsilon $$ of the above problem which exhibits $$l_0$$ -boundary peaks and $$l_1$$ -interior peaks for small $$\varepsilon >0$$ under rather strong conditions on f, such as the linearized non-degeneracy condition for a limiting problem. In this paper, we extend the previous result to more general class of f satisfying Berestycki–Lions conditions which we believe to be almost optimal.

1.1. Throughout this paper, let S be a fixed, smooth (C ~) orientable closed surface of genus g > 2. Let S be equipped with a preferred complex structure; we denote the Riemann surface byX0, and the Teichmtiller space ofX o by T 0. There is a Fuchsian group Fg operating in the unit disk so that X o = U/F o. The universal Teichmiiller curve V o is a fibre space over Teichmtiller space, with Xt, the fibre over tE TO conformally equivalent to the Riemann surface represented by te T 0. (Each X t is diffeomorphic to S.) V o is a complex manifold of dimension 3 9 2 , and we denote its canonical line bundle by K(V0). We define L, a holomorphic complex line bundle over Vo, to be an n th root of K(V0) iffL | K(V0). The purpose of this paper is to study the n ttt roots of K(V0); in particular, we investigate an action of the Teichmtiller modular group Mod(Fg) (= mapping class group) on the (finite) set of n th roots. Denote the tangent and cotangent bundles of a manifold M by T(M) and T*(M), respectively; To(M) and T~(M) denote the corresponding bundles with their zero sections removed. David Mumford observed (informal communication) that over a single Riemann surface X, the n th roots correspond to certain homomorphisms 2: HI(To(X), Z.)~Z.. He suggested trying to work out the action of Mod(F 0) in terms of its action on these homomorphisms. 1.2. Let f : S ~ S be a diffeomorphism, and ]" its equivalence class in Mod(Fg). Mod(Fo) acts on V 0 as a group of biholomorphic maps, and that action induces (by pullback) an action on the set of holomorphic complex line bundles which a r e n th roots of K(V0). The n tla roots are a finite set of order n 2g. The action of Mod(Fg) on that set gives a homomorphism Mod(Fg)--*Perm(n2~ where Perm(n 2g) is the permutation group on a set of order n 2g. Let Go, . denote the kernel of that homomorphism, that is, the subgroup of Mod(Fo) which acts trivially on all n tla roots. These groups are of particular interest, because they are normal subgroups of finite index in Mod(Fg). The study of the action of Mod(Fg) o n n th roots leads to the following characterization of these subgroups

j = l holomorphic mappings from B, into B k. We then apply this principle to the problem of characterizing the proper holomorphic mappings from B, into B k. Two such mappings f and g are said to be equivalent if there exist holomorphic automorphisms ~b and ~p of B, and Bk, respectively, such that f=~pogo49. There are at present three significant papers on the subject. The first by Alexander [1] proves that i f f is a proper holomorphic mapping of B, into B, then f is a holomorphic automorphism of B,. Webster I-5] proved that for n > 3, i f f is a proper holomorphic mapping of B, into B,+ 1 which is C3(/]) then f maps B, into a hyperplane in r Hence using Alexander's result and properties of the holomorphic automorphisms, f is equivalent to the mapping (zl ,z2, . . . ,z , ) "--~(ZI,Z2,...,Zn, O ). Alexander's example (zl ,z2)-~(z2,]/~zlz2,z 2) shows that Webster's result does not hold for n = 2. In a recent paper, Faran [2] has characterized all proper holomorphic mappings f from B 2 to B 3 that satisfy the additional smoothness condition feC3(/~2). He proves that such mappings are equivalent to one of the four mappings (z, w) --,(z, w, 0), (z, w)---~(z, zw, w2), (z, w)---r(z 2, l/~zw, w2)

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: $\mathbb{S}^4$, $\mathbb{CP}^2$, $\mathbb{S}^2\times\mathbb{S}^2$, or $\mathbb{CP}^2\#\pm \mathbb{CP}^2$. As an application we prove that any closed simply connected 4-manifold admitting a nontrivial singular Riemannian foliation is diffeomorphic to a connected sum of copies of standard $\mathbb{S}^4$, $\pm\mathbb{CP}^2$ and $\mathbb{S}^2\times\mathbb{S}^2$. A classification of singular Riemannian foliations of codimension 1 on all closed simply connected 4-manifolds is obtained as a byproduct. In particular, there are exactly 3 non-homogeneous singular Riemannian foliations of codimension 1, complementing the list of cohomogeneity one 4-manifolds., Comment: 24 pages, final version, to appear in Math. Ann

In this paper, we study the Ricci flat manifolds with maximal volume growth using Perelman’s reduced volume of Ricci flow. We show that if \((M^n,g)\) is an noncompact complete Ricci flat manifold with maximal volume growth satisfying \(|Rm|(x)\rightarrow 0\) as \(d(x)=d_g(x,p)\rightarrow \infty \), then \(M^n\) has the quadratic curvature decay. Some applications to this result are also presented.

In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ for $n\ge 6$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2, Comment: 18 pages

We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation, $$\partial_t u=\Delta u+F(x,t,u) \quad{\rm in} \;{\bf R}^N\times(0,\infty), \quad u(x,0)=\varphi(x)\quad{\rm in} \;{\bf R}^N,\quad\quad ({\rm P})$$ where \({F\in C({\bf R}^N\times[0,\infty)\times{\bf R})}\) and \({\varphi\in L^1({\bf R}^N, (1+|x|)^K\,dx)}\) with K ≥ 0. Assume that u is a solution of (P) satisfying $$|F(x,t,u(x,t))|\le C(1+t)^{-A}|u(x,t)|,\quad (x,t)\in{\bf R}^N\times(0,\infty),$$ for some constants C > 0 and A > 1. Then it is well known that the solution u behaves like the heat kernel. In this paper we give the ([K] + 2)th order asymptotic expansion of the solution u, and reveal the relationship between the asymptotic profile of the solution u and the nonlinear term F. Here [K] is the integer satisfying K − 1 < [K] ≤ K.

In this paper, we investigate the closure of a large class of Teichmuller discs in the stratum \({\mathcal{Q}(1, 1, 1, 1)}\) or equivalently, in a \({{\rm GL}^+_2(\mathbb{R})}\) -invariant locus \({\mathcal{L}}\) of translation surfaces of genus three. We describe a systematic way to prove that the \({{\rm GL}^+_2(\mathbb{R})}\) -orbit closure of a translation surface in \({\mathcal{L}}\) is the whole locus \({\mathcal{L}}\) . The strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of Ratner’s theorem to unipotent subgroups acting on an “adequate” splitting. This analysis applies for example to all Teichmuller discs obtained by the Thurston–Veech’s construction with a trace field of degree three which are moreover “obviously not Veech”. We produce an infinite series of such examples and show moreover that the favourable splitting situation does not arise everywhere on \({\mathcal{L}}\) , contrary to the situation in genus two. We also study completely periodic directions on translation surfaces in \({\mathcal{L}}\) . For instance, we prove that completely periodic directions are dense on surfaces obtained by the Thurston–Veech’s construction.

The present paper is concerned with the Cauchy problem $$\left\{\begin{array}{ll}\partial_t u = \Delta u^m + u^p & \quad {\rm in}\; \mathbb R^N \times (0,\infty),\\ u(x,0) = u_0(x) \geq 0 & \quad {\rm in}\; \mathbb R^N, \end{array}\right.$$ with p, m > 1. A solution u with bounded initial data is said to blow up at a finite time T if \({{\lim {\rm sup}_{t \nearrow T}||u(t)||_{L^\infty(\mathbb{R}^N)} =\infty}}\). For N ≥ 3 we obtain, in a certain range of values of p, weak solutions which blow up at several times and become bounded in intervals between these blow-up times. We also prove a result of a more technical nature: proper solutions are weak solutions up to the complete blow-up time.

The general aim of this paper is to study which are the solvable Lie groups admitting an Einstein left invariant metric. The space N of all nilpotent Lie brackets on R^n parametrizes a set of (n+1)-dimensional rank-one solvmanifolds, containing the set of all those which are Einstein in that dimension. The moment map for the natural GL(n)-action on N evaluated in a point of N encodes geometric information on the corresponding solvmanifold, allowing us to use strong and well-known results from geometric invariant theory. For instance, the functional on N whose critical points are precisely the Einstein solvmanifolds is the square norm of this moment map. We also use a GL(n)-invariant stratification for the space N following essentially a construction given by F. Kirwan and show that there is a strong interplay between the strata and the Einstein condition on the solvmanifolds. As applications, we obtain several examples of graded (even 2-step) nilpotent Lie algebras which are not the nilradicals of any standard Einstein solvmanifold, as well as a classification in the 7-dimensional 6-step case and an existence result for certain 2-step algebras associated to graphs., Comment: Final version to appear in Math. Annalen

In earlier papers we studied direct limits $${(G,\,K) = \varinjlim\, (G_n,K_n)}$$ of two types of Gelfand pairs. The first type was that in which the G n /K n are compact Riemannian symmetric spaces. The second type was that in which $${G_n = N_n\rtimes K_n}$$ with N n nilpotent, in other words pairs (G n , K n ) for which G n /K n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections $${\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}$$ for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit $${L^2(G/K) := \varinjlim L^2(G_n/K_n)}$$ is multiplicity-free. This left open questions concerning the nature of the elements of L 2(G/K). Here we define spaces $${\mathcal{A}(G_n/K_n)}$$ of regular functions on G n /K n and injections $${\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}$$ for m ≧ n related to restriction by $${\nu_{m,n}(f)|_{G_n/K_n} = f}$$ . Thus the direct limit $${\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}$$ sits as a particular G-submodule of the much larger inverse limit $${\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}$$ . Further, we define a pre Hilbert space structure on $${\mathcal{A}(G/K)}$$ derived from that of L 2(G/K). This allows an interpretation of L 2(G/K) as the Hilbert space completion of the concretely defined function space $${\mathcal{A}(G/K)}$$ , and also defines a G-invariant inner product on $${\mathcal{A}(G/K)}$$ for which the left regular representation of G is multiplicity-free.

We consider Levi non-degenerate tube hypersurfaces in \({\mathbb{C}^{n+1}}\) that are (k, n − k)-spherical, i.e. locally CR-equivalent to the hyperquadric with Levi form of signature (k, n − k), with n ≤ 2k. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) k = n − 2, n ≥ 7; (ii) k = n − 3, n ≥ 7; (iii) k ≤ n − 4. For all other values of k and n, except for k = 3, n = 6, the number of affine classes was known to be finite. The exceptional case k = 3, n = 6 has been recently resolved by Fels and Kaup who gave an example of a family of (3, 3)-spherical tube hypersurfaces that contains uncountably many pairwise affinely non-equivalent elements. In this paper we deal with the Fels–Kaup example by different methods. We give a direct proof of the sphericity of the hypersurfaces in the Fels–Kaup family, and use the j-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.

Let \({k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})}\) . The single valued real analytic n-polylogarithm \({\mathcal{L}_{n}: \mathbb{C} \to \mathbb{R}}\) is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in Unver (Algebra Number Theory 3:1–34, 2009) to define additive n-polylogarithms \({li_{n}:k[\varepsilon]_{2}\to k}\) and prove that they satisfy functional equations analogous to those of \({\mathcal{L}_{n}}\). Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group \({B_{n}' (k[\varepsilon]_{2})}\) defined by Goncharov (Adv Math 114:197–318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over k[e]2.

Let f be a holomorphic cusp form of weight l on SL2(Z) and Ω an algebraic Hecke character of an imaginary quadratic field K with Ω((α)) = (α/|α|)l for \({\alpha\in K^{\times}}\). Let L(f, Ω; s) be the Rankin-Selberg L-function attached to (f, Ω) and P(f, Ω) an “Ω-averaged” sum of CM values of f. In this paper, we give a formula expressing the central L-values L(f, Ω; 1/2) in terms of the square of P(f, Ω).

Inspired by Riemann’s work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form $${\frac{\eta(A_jz)} {\eta(A_{j-1}z)}}$$ , where A j-1 and A j are matrices whose rows are the coordinates of consecutive visible lattice points in a dilation XΩ of a fixed region Ω in $${\mathbb{R}^2}$$ , and z is a fixed complex number in the upper half plane. In particular, we show that the limiting distribution of these quotients depends heavily on the index of Farey fractions which was first introduced and studied by Hall and Shiu. The distribution of Farey fractions with respect to the value of the index dictates the universal limiting behavior of these quotients. Motivated by chains of these quotients, we show how to obtain a generalization, due to Zagier, of an important formula of Hall and Shiu on the sum of the index of Farey fractions.

Let $$M^\circ$$ be a complete noncompact manifold and g an asymptotically conic manifold on $$M^\circ$$ , in the sense that $$M^\circ$$ compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M. A special case that we focus on is that of asymptotically Euclidean manifolds, where the induced metric at infinity is equal to the standard metric on S n−1; such manifolds have an end that can be identified with $${\mathbb{R}}^n \backslash B(R,0)$$ in such a way that the metric is asymptotic in a precise sense to the flat Euclidean metric. We analyze the asymptotics of the resolvent kernel (P + k 2)−1 where $$P = \Delta_g + V$$ is the sum of the positive Laplacian associated to g and a real potential function $$V\in C^{\infty}(M)$$ which vanishes to second order at the boundary (i.e. decays to second order at infinity on $$M^\circ$$ ) and such that $$\Delta_{\partial M}+(n-2)^2/4+V_0 > 0$$ if $$V_0:=(x^{-2}V)|_{\partial M}$$ . Then we show that on a blown up version of $$M^2 \times [0, k_0]$$ the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary, and we are able to identify explicitly the leading behaviour of the kernel at each boundary hypersurface. Using this we show that the Riesz transform of P is bounded on $$L^p(M^\circ)$$ for 1 n depends explicitly on the first non-zero eigenvalue of the Laplacian on the boundary $$\partial M$$ . Our results hold for all dimensions ≥ 3 under the assumption that P has neither zero modes nor a zero-resonance. In the follow-up paper Guillarmou and Hassell (Resolvent at low energy and Riesz transform for Schrodinger operators on asymptotically conic manifolds, preprint) [7] we analyze the same situation in the presence of zero modes and zero-resonances.

This paper deals with nonnegative solutions of for $$u_t = \Delta u - u^{-q} \chi_{\{u > 0\}} \qquad {\rm in}\,\Omega \times (0,\infty) \qquad \qquad ({\rm Q})$$ with \(q \in (0, 1)\) and prescribed continuous Dirichlet data B = B(x) on ∂Ω. It is proved that for n ≤ 6 there is a critical parameter\(q_c \in (0, 1)\) with the following property: If q > qc then there exist at least two continuous weak solutions emanating from some explicitly known stationary solution w: one that coincides with w and another one that satisfies u ≥ w but \(u \not\equiv w\) . For n ≤ 6 and q ≤ qc (or n ≥ 7), however, such a second solution above w is impossible. Moreover, it is shown that for n ≤ 6, q > qc and any sufficiently small nonnegative boundary data B there exist initial values admitting at least two continuous weak solutions of (Q). The final result asserts that for any n and q nonuniqueness for (Q) holds at least for some boundary and initial data.

We introduce the concept of average homogeneity of a measure by comparing the measure to the uniform distribution in a relatively simple way. This leads to a very general notion which may be regarded as an inverse of porosity. In this paper the emphasis is given to relations between homogeneity and dimensions of measures. First we consider the effect of homogeneity on dimensions by proving an upper bound to the Hausdorff dimension as a function of homogeneity and its order. The opposite question of how dimensions effect homogeneity is solved by giving an upper bound to homogeneity in terms of upper packing dimension. We also illustrate by examples that all our results are the best possible ones.

This paper is concerned with where Ω is a bounded smooth domain in R N , T D >0, D>0, and p>1 with (N−2)p≤N+2. Let P 2 be the projection from L 2(Ω) onto the second Neumann eigenspace. We prove that, if P 2φ≠0 in Ω and D is sufficiently large, the solution u of (P) blows up only near the set ℳ∪∂Ω, where ${{{{\cal{ M}}}=\{x\in\overline{{\Omega}}:(P_2 \phi)(x) =\max_{{y\in\overline{{\Omega}}}}(P_2 \phi)(y)\}}}$ .

In this paper we investigate the blowup criteria of the quasilinear parabolic system \({{ u_{{\imath t}}=c_{{\imath}}u_{{\imath}}^{{\alpha_{{\imath}}}}(\Delta u_\imath+ \prod_{{\jmath=1}}^n u_\jmath^{{p_{{\imath\jmath}}}}), \imath=1, 2, \cdots, n }}\) with homogeneous Dirichlet boundary conditions on a bounded domain Ω⊂RN, where ci>0, αi>0, pi ≥0 (1≤i, ≤n) are constants. Denote by I the identity matrix and P=(pi ), which is assumed to be irreducible. That I−P is a singular M-matrix is shown to be the critical case, in which λ1 plays a fundamental role, where λ1 is the first Dirichlet eigenvalue of the Laplacian on Ω. As a result, we give a general answer to the question of Galaktionov and Levine on the porous medium systems.

The affine homogeneous space \(G^{\Bbb C}/K^{\Bbb C}\) associated to a real semi-simple Lie group G with maximal compact subgroup K contains a number of naturally defined{\it G}-invariant neighborhoods of its real points \(M_{\Bbb R} = G/K\) which are of interest from various points of view. Here the universal Iwasawa domain \(\Omega_I\) is introduced from the point of view of incidence geometry and certain of its properties are derived, e.g., it is Stein, Kobayashi hyperbolic and contains the domain \(\Omega_{AG}\) introduced by Akhiezer and Gindikin which is now known to be equivalent to the maximal domain of definition \(\Omega_{adp}\) of the adapted complex structure associated to the Killing metric in the tangent bundle \(TM_{\Bbb R}\). One of the main goals of the paper is to develop methods which lead to a better understanding of the Wolf domain \(\Omega_W(D)\) of cycles in an open G-orbit D in a flag manifold \(G^\Bbb C/P\). The key is the Schubert domain \(\Omega_S(D)\) which is defined by Schubert cycles of complementary dimension to the cycles. These are defined by a Borel subgroup containing an Iwasawa factor AN and consequently \(\Omega_S(D)\) and \(\Omega_I\) are closely related.

This paper characterizes the subsets E of \(\mathbb{R}^n\) which have the following property: there exists a harmonic function u on \(\mathbb{R}^n\times(0,+\infty)\) such that \(u(x,t) \to +\infty\) as \(t \to 0+\) for each x inE. This problem has its roots in classical work of Lusin and Privalov, and the answer has been known for some time in the case where n=1. However, the characterization turns out to be more delicate in higher dimensions.

Let \( M^m\) be an umbilic-free submanifold in \(S^n\) with I and II as the first and second fundamental forms. An important Moebius invariant for \(M^m\) in Moebius differential geometry is the so-called Moebius form \(\Phi\), defined by \(\Phi =-\rho^{-2}\sum_{i,\alpha}\{H^{\alpha}_{,i} +\sum_j(II^{\alpha}_{ij}-H^{\alpha}I_{ij})e_j(\log \rho)\{\omega_i\otimes e_{\alpha}\), where \(\{e_i\}\) is a local basis of the tangent bundle with dual basis \(\), \(\{e_{\alpha}\}\) is a local basis of the normal bundle, \(H=\sum_{\alpha}H^{\alpha}e_{\alpha}\) is the mean curvature vector and \(\rho =\sqrt{m\over{m-1}}\|II-HI\|\). In this paper we prove that if \(x: S^2\to S^n\) is an umbilics-free immersion of 2-sphere with vanishing Moebius form \(\Phi\), then there exists a Moebius transformation \(\tau: S^n\to S^n\) and a 2k-equator \(S^{2k}\subset S^n\) with \(2\le k\le [n/2]\) such that \(\tau\circ x:S^2\to S^{2k}\) is the Veronese surface.

Let V be a Euclidean Jordan algebra, and let $\Omega$ be the corresponding symmetric cone. The geometric mean $a\#b$ of two elements a and b in $\Omega$ is defined as a unique solution, which belongs to $\Omega,$ of the quadratic equation $P(x)a^{-1}=b,$ where P is the quadratic representation of V. In this paper, we show that for any a in $\Omega$ the sequence of $n^{\mathrm{th}}$ iterate $f_a^n(x)$ of the function $f_a:\Omega\to \Omega$ defined by $ f_a(x)={1\over 2}(x+P(a)x^{-1})$ converges to a. As applications, we obtain that the geometric mean $a\#b$ of $\Omega$ can be represented as a limit of successive iteration of arithmetic means and harmonic means, and we derive the Lowner-Heinz inequality on the symmetric cone $\Omega: 0\leq a\leq b {\mathrm{ implies}} a^p\leq b^p{\mathrm{ for}} 0\leq p\leq 1.$ Furthermore, we obtain a formula ${\mathrm exp} (x+y)=\lim_{n\to \infty}({\mathrm exp}{\frac{2}{n}{x}}\#\exp{\frac{2}{n}}y)^n$ which leads a Golden-Thompson type inequality $||{mathrm exp} (x+y)||\leq ||P({\mathrm exp}{x\over 2}){\mathrm exp} y||$ for the spectral norm on V.

This paper establishes and extends a conjecture posed by M. Gromov which states that every riemannian metric $g$ on $S^n$ that strictly dominates the standard metric $g_0$ must have somewhere scalar curvature strictly less than that of $g_0$ . More generally, if $M$ is any compact spin manifold of dimension $n$ which admits a distance decreasing map $f:M \rightarrow S^n$ of non-zero degree, then either there is a point $x \in M$ with normalized scalar curvature $\tilde{\kappa}(x)< 1$ , or $M$ is isometric to $S^n$ . The distance decreasing hypothesis can be replaced by the weaker assumption $f$ is contracting on $2$ -forms. In both cases, the results are sharp. An explicit counterexample is given to show that the result is no longer valid if one replaces 2-forms by $k$ -forms with $k \geq 3$ .

In this paper, we consider indefinite sequences having singularity at the point ( = (1 . . . . . 1) ~ (I~ p of order at most 2x. We show that they are related to both the classical moment problem and the L6vy-Khinchin formula for negative definite sequences. We establish an integral representation for these sequences and characterize the exponentially bounded ones by means of finitely many positive definiteness conditions. Even in the case x = 1, our approach provides new L6vy-Khinchin type integral representations. Let No be the set of all non-negative integers and fix the positive integer p. We denote by ~ l -z , i ] the algebra of polynomials in z and ~ where z = (zl . . . . . zp ) e r p. This algebra acts on the sequences ct = ~(m, n), (m,n)e ]Ng • N~ following the correspondence

In Chang and Yang 1,5], a type of Sobolev inequality was proved for a class of functions on S 2. It played an important role in the proof of the existence, by the min-max method, of a pointwise conformal metric on S 2 which has some given function as its scalar curvature. In this paper, we prove an analogous inequality for functions on S n (n > 3). From now on, ~ denotes the integral average on S", i.e., ~ f = (~s . fdx) /vol(S") for functions on S", and x j ( j = 1, 2 . . . . . n + 1) denote orthogonal coordinates in R "+1.

where c e R, 0 < c < 1, and %2 = \Q4 Q4/ v Therefore, p. 951'8, ".. .are involutions of S3/H '' should read as ".. .are of order 2 or 3 as an element of the diffeomorphism group of S3/H ''. The error was in (41). In case K = B2, the possibility of(detg)1/2g = vn~ was missing. This error had its origin in that of Lemma 4 in my previous paper "Topology of Hopf surfaces", J. Math. Soc. Japan 27, 222-238 (1975). The correction of that paper will appear in J. Math. Soc. Japan.

In this paper we continue our investigation initiated in ECN] on the structure of the set of all possible solutions of the equation (1.1) Au+ Ke2"=O in R 2, where A = is the Laplace operator and K is a given nonpositive, locally HSlder continuous function in R 2. Equation (1.1) arises in the problem of finding a Riemannian metric which realizes the given function K as its Gaussian curvature and is conformal to the standard Euclidean metric in R 2. We refer the reader to [CN] for a brief description of the background and the history of this problem. The study of the equation (1.1) dates back to at least 1930's but it seems that a complete classification of all solutions of (1.1) was obtained only very recently in [CN] for the case K(x) ~ Ix[t near oo, where l > 2 is a constant. (Here we continue to use the notation '~',-~g near ~ " to denote that "there exist two positive constants Cl, C2 such that C l f > g > C2f near oo" as we did in [CN].) More precisely, the following result was established in [CN]. Theorem. Suppose that K 2. Then the following conclusions hold. (i) For each ~ 0 , ~ , (1.1) possesses a unique solution u~ such that

(1.1) Let c be a real number. Represent by A4"(c) a n-dimensional space form of curvature c. Let M N be a N-dimensional connected Riemannian manifold. The question that served as the starting point for this paper was to find simple conditions on the metric of M so that, if f:M"~ffl"+P(c), p>= 1, is an isometric minimal immersion then f is rigid in the following sense: given another minimal immersion g: M" ~ A4" + q(c), q >= p, then there exists a rigid motion T of M~ + q(c) such that g = To f, (~l"+P(c) being considered as a totally geodesic submanifold of

Let X and Y be locally convex Hausdorff spaces and L(X, Y) the space of continuous linear operators from X to Y. If TeL(X, Y), we denote by N(T) its kernel or null space and by R(T) its range, q~-(X, Y) is defined as the set of all operators TeL(X, Y) which are open, have a closed range and finite nullity nul(T) := dimN(T). Mappings of this type are called lower semi-Fredholm or ~ operators, q)+(X, Y) is defined as the set of those operators TeL(X, Y) which are open and have a closed range of finite codimension, i.e. they have deficiencies clef(T) := codim R(T)< oo. Operators of this kind are called upper semi-Fredholm or q~+-operators. If T is a lower or upper semi-Fredholm operator, then the index of T is defined as ind(T):=nul(T)-def(T). An operator which belongs to 9 (X, Y):= ~-(X, Y)c~+(X, Y) is said to be a Fredholm or q~-operator. We denote by ~Z(X, Y) or ~r(X, Y) the subset of those operators in ~-(X, Y) or ~+(X, Y) respectively which have projectable kernels and ranges; these operators are called ~,. or q~r-operators. In this present paper G is a region in C N. We consider H(G, L, (X, Y)), the family of holomorphic maps from G to L(X, Y), where L(X, Y) is provided with the topology of pointwise convergence. Let H(G, q~• Y)) be the subset of those functions in H(G, L(X, Y)) which have values in ~+(X, Y) or ~-(X, Y) respectively. We consider the set of sinoularities

The conclusion "obtained from a Stein space by blowing up finitely many points" means precisely that there exist: a compact analytic set S C X with d imxS>0 for any x ~ S; a Stein space 14, a finite set A C Y and a proper holomorphic map p : X ~ Y inducing a biholomorphism X\S ~ Y\A and which satisfies p, C!~x ~ Cr~r. Following a customary terminology X is called an 1-convex space, S its exceptional set and Y the Remmert reduction of X. Theorem 1.1. was conjectured by Fornaess-Narashimhan in [5]. In this paper we prove the converse

O. Introduction Let A be a uniform algebra on a compact subset K of a complex space X such that A contains a subset F of function holomorphic in an open neighborhood Y of K such that F separates points in Y. It is shown that the Shilov boundary of A is not almost thin in K. As an example, let G ~ 0 be an open and bounded subset of ~n and consider the uniform algebra of functions which are uniform limits on of polynomials. Then there exists a unique minimal closed subset S(A) of G such that for every function f in A, the maximum of Ifl is taken on S(A). The set S(A) is contained in the boundary of G and called the Shilov boundary. Let U be the unit disc and let T be the unit circle. In the case G = U n is the polydisc, the Shilov boundary is the torus S(A) = T n. Here S(A) is small compared with G or its boundary dG = G-G. The topological dimension of S(A) is half that of G in this example. On the other hand, if G is a ball or any open, bounded subset of C" having a smooth, strongly pseudo-convex boundary OG, then $(A)= OG. These two examples make it seem that the size of S(A) varies from extremely small to as large as possible. The purpose of this paper is to demonstrate that, from a holomorphic point of view, S(A) and G have exactly the same size. Essentially the same fact is also true when G is taken to be any compact subset K of a complex space X and if A is any uniform algebra on K which contains sufficiently many holomorphic functions. These results simplify some proofs in [6].

It splits over the maximal compact subgroup. Let T be the corresponding lift of I, the Iwahori subgroup of G. Now fix ~, a faithful character of #n(F). Let R}(~) be the category of smooth representations of ~ which are generated by its T fixed vectors and such that/~n(F) acts via ~. Also, let H~(~) be the Hecke algebra of I" biinvariant, compactly supported functions on ~ satisfying f(gQ = f(g)~(~ For (z, ~, V) ~ Obj R}(~) let V ~ be the space of I'-fixed vectors. Then f ~ ( f ) l V ~ is a representation of H~(t~). According to Bernstein and Borel [2] this gives the equivalence of R}(~) and the category of representations of H~(~). In particular, this is the case for G, whose Hecke algebra will be denoted by H(G) and category by RI(G). In this paper we describe H~(~) in terms of generators and relations. As a consequence, in the case of simple group, for example, we find a group G' related to G such that H*(t~)~ H(G'). In particular, we get the equivalence of R}(t~) and

In this paper we exclusively deal with a connected closed complex submanifold V of dimension n > 1 in a complex projective s p a c e / ~ of dimension m which is a complete intersection. Namely, V is assumed to admit r = m n homogeneous polynomials P1 . . . . . P, on C "+ 1, which not only define V as their zero locus but also generate the ideal associated to V. The purpose is to prove the following

The problem of density of smooth maps between manifolds was posed in [EL] and in [SU]. In [BZ], Bethuel and Zheng showed that smooth maps from an open subset 12 of R ~ are dense in WI'P(O, S ~ 1) if 1 < p < M 1. Here we extend this result to approximate maps of W 1' P(f2, S M1) which have some prescribed smooth boundary data by maps with the same boundary data, and which are smooth on fl excepted on a closed subset 6 a with zero p-capacity, still if 1 ~ p < M 1. The first part of this paper is devoted to the basic tools used in the proof of this result. In the second part the proof of our main theorem is given; more precisely: We consider p a real number in [1, + ~ ) and two integer N and M larger than 1. S ~ 1 = {y ~ RM/Iy] = 1 } is the unit sphere of R M. 12 is a bounded open set of R N, whose boundary dr2 is a smooth (N-1)-dimensional compact manifold. Let f be in WI'P(f2,SU-1), we define

Let D be bounded symmetric domain of rank p, and let o9 be the KRhler form of its Bergmann metric, where the metric is normalized so that the minimum holomorphic sectional curvature is 1. I fX is a compact manifold whose universal cover is D, to defines a cohomology class [to] e H 2 (X, R). The purpose of this paper is to compute its sup norm in the sense of Gromov [5]. We show II [ to] II co = pn. Strictly speaking, we only prove this for three of the four classes of classical domains (cf. Sect. 2). This theorem has the following topological corollary. Let S be a Riemann surface of genus 0 > 1 and f : S ~ X a continuous map. Then ! f ' t o < 4 p ( # 1)n.

in this note we consider smooth rational curves C of degree n in threedimensional projective space IP 3 (over a closed field of characteristic 0). To avoid trivial exceptions we shall always assume that n ~ 4 (this does not hold however for certain auxiliary curves we shall consider). Let N = N c be the normal bundle of C in IP 3. Since degel(IP3)=4, and d e g c l ( l P 0 = 2 , we have that d e g c l ( N ) = 4 n 2 . By a well-known theorem of Grothendieck the bundle N is a direct sum of two line bundles. Hence N ~ O c ( 2 n l a ) G O c ( 2 n 1 +a) for some non-negative a=a(C), which is uniquely determined by C. The question we would like to answer is an obvious one: which values of a occur? We shall show (Theorem 4 below) that a value a occurs if and only if 0_ =0, therefore Hi(C, N)=O. It follows [K, p. 150] that C represents a smooth point on the Chow variety Ch(3, 1, n) of effective cycles of dimension 1 and degree n in IP 3. Since the set of all smooth rational curves with a fixed degree is obviously connected, we see that the smooth C's represent a smooth, irreducible, 4n-dimensional (Zariski-)open subset S of Ch(3, 1, n). In a for thcoming paper [ E V ] we shall prove the following

Let M be a submanifold of a Riemannian manifold ]fir and P an endomorphism of the tangent bundle T M of M. If F T x M 3_T ,M for each point x¢ M, we say that M is an anti-invariant submanifold of M under F. If P is an almost complex structure, then such M are usually called totally real submanifolds and have been studied by various authors [1, 3, 5-7]. The purpose of the present paper is to study the case, in which P is the endomorphism q) of an almost contact metric structure (~p, ~, r/, 9) on AT1, in particular, that of a Sasakian structure (see § 2). The authors will study the cases in which the structure vector field ~ is either tangent to M (see § 4) or normal to M (see § 5). In both cases the computation of the Laplacian of the square of the norm of the second fundamental form of M in AS1 plays an important role. The Theorems 8, 15, 16 present typical examples of our main results, which say, roughly speaking, that compact, minimal, anti-invariant submanifolds M of a Sasakian space form ~/, which are of not too large relative curvature (measured in terms of the norm I[cq[ of the second fundamental form ~ of M in .g/) have to be already totally geodesic in hT/.

Let X be a complete non-singular algebraic curve of genus 9 > 2 over an algebraically closed field k, n a positive integer, d an integer and L a line bundle of degree d over X. It is well known that the isomorphism classes of stable bundles of rank n and determinant L over X form an irreducible non-singular quasiprojective variety S,,L(X) of dimension (n 2 1 ) (9-1) , which is projective if (n,d)= 1 (see [8, 10-13]). It is also easy to see that S,,L(X) is unirational. Our object in this paper is to prove