Asian Journal of Mathematics Articles (Project Euclid)
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The latest articles from Asian Journal of Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTSat, 28 May 2011 16:54 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Cross Curvature Flow on Locally Homogeneous Three-manifolds (II)
http://projecteuclid.org/euclid.ajm/1275671452
<strong>Xiaodong Cao</strong>, <strong>Laurent Saloff-Coste</strong><p><strong>Source: </strong>Asian J. Math., Volume 13, Number 4, 421--458.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the positive cross curvature flow on locally homogeneous
3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results
concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided
behavior of maximal solutions of the cross curvature flow on locally homogeneous 3-manifolds. We
show that, typically, the positive cross curvature flow on locally homogeneous 3-manifold produce
an Heisenberg type sub-Riemannian geometry.
</p>projecteuclid.org/euclid.ajm/1275671452_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTP. D. E.'s Which Imply the Penrose Conjecturehttp://projecteuclid.org/euclid.ajm/1331583349<strong>Hubert L. Bray</strong>, <strong>Marcus A. Khuri</strong><p><strong>Source: </strong>Asian J. Math., Volume 15, Number 4, 557--610.</p><p><strong>Abstract:</strong><br/>
In this paper, we show how to reduce the Penrose conjecture to the known Riemannian
Penrose inequality case whenever certain geometrically motivated systems of equations can be
solved. Whether or not these special systems of equations have general existence theories is therefore
an important open problem. The key tool in our method is the derivation of a new identity which we
call the generalized Schoen-Yau identity, which is of independent interest. Using a generalized Jang
equation, we use this identity to propose canonical embeddings of Cauchy data into corresponding
static spacetimes. In addition, we discuss the Carrasco-Mars counterexample to the Penrose conjecture
for generalized apparent horizons (added since the first version of this paper was posted on the
arXiv) and instead conjecture the Penrose inequality for time-independent apparent horizons, which
we define.
</p>projecteuclid.org/euclid.ajm/1331583349_Mon, 12 Mar 2012 16:15 EDTMon, 12 Mar 2012 16:15 EDTA New Pinching Theorem for Closed Hypersurfaces with Constant Mean Curvature in $S^{n+1}$http://projecteuclid.org/euclid.ajm/1331583350<strong>Hong-Wei Xu</strong>, <strong>Ling Tian</strong><p><strong>Source: </strong>Asian J. Math., Volume 15, Number 4, 611--630.</p><p><strong>Abstract:</strong><br/>
We investigate the generalized Chern conjecture, and prove that if $M$ is a closed
hypersurface in $S^{n+1}$ with constant scalar curvature and constant mean curvature, then there exists
an explicit positive constant $C(n)$ depending only on $n$ such that if $|H| < C(n)$ and $S > \beta
(n,H)$, then $S > \beta (n,H) + \frac{3n}{7}$, where $\beta(n,H) = n + \frac{n^3 H^2}{2(n−1)} + \frac{n(n−2)}{2(n−1)} \sqrt{n^2 H^4 + 4(n − 1)H^2}$.
</p>projecteuclid.org/euclid.ajm/1331583350_Mon, 12 Mar 2012 16:15 EDTMon, 12 Mar 2012 16:15 EDTOn the Affine Homogeneity of Algebraic Hypersurfaces Arising from Goernstein Algebrashttp://projecteuclid.org/euclid.ajm/1331583351<strong>A. V. Isaev</strong><p><strong>Source: </strong>Asian J. Math., Volume 15, Number 4, 631--640.</p><p><strong>Abstract:</strong><br/>
To every Gorenstein algebra $A$ of finite vector space dimension greater than 1
over a field $\mathbb{F}$ of characteristic zero, and a linear projection $\pi$ on its maximal ideal $\mathfrak{m}$ with range
equal to the annihilator $\operatorname{Ann}(\mathfrak{m})$ of $\mathfrak{m}$, one can associate a certain algebraic hypersurface $S_{\pi} \subset \mathfrak{m}$.
Such hypersurfaces possess remarkable properties. They can be used, for instance, to help
decide whether two given Gorenstein algebras are isomorphic, which for $\mathbb{F} = \mathbb{C}$ leads to interesting
consequences in singularity theory. Also, for $\mathbb{F} = \mathbb{R}$ such hypersurfaces naturally arise in CR-geometry.
Applications of these hypersurfaces to problems in algebra and geometry are particularly
striking when the hypersurfaces are affine homogeneous. In the present paper we establish a criterion
for the affine homogeneity of $S_{\pi}$ . This criterion requires the automorphism group $\operatorname{Aut}(\mathfrak{m})$ of $\mathfrak{m}$ to
act transitively on the set of hyperplanes in m complementary to $\operatorname{Ann}(\mathfrak{m})$. As a consequence of this
result we obtain the affine homogeneity of $S_{\pi}$ under the assumption that the algebra $A$ is graded.
</p>projecteuclid.org/euclid.ajm/1331583351_Mon, 12 Mar 2012 16:15 EDTMon, 12 Mar 2012 16:15 EDTLagrangian unknottedness in Stein surfaceshttp://projecteuclid.org/euclid.ajm/1331663450<strong>Richard Hind</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 1, 1--36.</p><p><strong>Abstract:</strong><br/>
We show that the space of Lagrangian spheres inside the cotangent bundle of the
2-sphere is contractible. We then discuss the phenomenon of Lagrangian unknottedness in other
Stein surfaces. There exist homotopic Lagrangian spheres which are not Hamiltonian isotopic, but
we show that in a typical case all such spheres are still equivalent under a symplectomorphism.
</p>projecteuclid.org/euclid.ajm/1331663450_Tue, 13 Mar 2012 14:30 EDTTue, 13 Mar 2012 14:30 EDTCosmological time versus CMC time in spacetimes of constant curvaturehttp://projecteuclid.org/euclid.ajm/1331663451<strong>Lars Andersson</strong>, <strong>Thierry Barbot</strong>, <strong>François Béguin</strong>, <strong>Abdelghani Zeghib</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 1, 37--88.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate the existence of foliations by constant mean curvature
(CMC) hypersurfaces in maximal, globally hyperbolic, spatially compact, spacetimes of constant
curvature.
In the non-positive curvature case (i.e. for flat and locally anti-de Sitter spacetimes), we prove
the existence of a global foliation of the spacetime by CMC Cauchy hypersurfaces. The positive
curvature case (i.e. locally de Sitter spacetimes) is more delicate: in general, we are only able to
prove the existence of a foliation by CMC Cauchy hypersurfaces in a neighbourhood of the past (or
future) singularity.
Except in some exceptional and elementary cases, the leaves of the foliation we construct are the
level sets of a time function, and the mean curvature of the leaves increases with time. In this case,
we say that the spacetime admits a CMC time function .
Our proof is based on using the level sets of the cosmological time function as barriers. A major
part of the work consists of proving the required curvature estimates for these level sets. One of
the difficulties is the fact that the local behaviour of the cosmological time function near one point
depends on the global geometry of the spacetime.
</p>projecteuclid.org/euclid.ajm/1331663451_Tue, 13 Mar 2012 14:30 EDTTue, 13 Mar 2012 14:30 EDTCodes from infinitely near pointshttp://projecteuclid.org/euclid.ajm/1331663452<strong>Bruce M. Bennett</strong>, <strong>Hing Sun Luk</strong>, <strong>Stephen S.-T. Yau</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 1, 89--102.</p><p><strong>Abstract:</strong><br/>
We introduce a new class of nonlinear algebraic-geometry codes based on evaluation of functions on inﬁnitely near points. Let $X$ be an algebraic variety over the ﬁnite ﬁeld $\mathbf{F}_q$. An inﬁnitely near point of
order $\mu$ is a point $P$ on a variety $X^\prime$ obtained by $\mu$ iterated blowing-ups starting from $X$. Given such a point $P$and a function $f$ on $X$, we give a deﬁnition of $f(P)$ which is nonlinear
in $f$ (unless $\mu = 0$). Given a set $S$ of inﬁnitely near points $\left\{P_1, \ldots , P_n \right\}$, we associate to $f$ its set of values $(f(P_1), \ldots, f(P_n))$ in $\mathbf{F}^n_q$. Let $V$ be a $k$ dimensional
vector space of functions on $X$. Evaluation of functions in $V$ at the $n$ points of $S$ gives a map $V \to \mathbf{F}^n_q$, which we view as an ($n, q^k, d$) code when the map is injective. Here d is the largest
integer such that a function in $V$ is uniquely determined by its values on any$n − d + 1$ points of $\mathcal{S}$. These codes generalize the Reed-Solomon codes, but unlike the $R-S$ codes they can be constructed
to have arbitrarily large code length $n$. The ﬁrst nontrivial case is where $X = A^2_{F_q}$, affine 2-space, and we study this case in detail.
</p>projecteuclid.org/euclid.ajm/1331663452_Tue, 13 Mar 2012 14:30 EDTTue, 13 Mar 2012 14:30 EDTHigher Bers mapshttp://projecteuclid.org/euclid.ajm/1331663453<strong>Guy Buss</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 1, 103--140.</p><p><strong>Abstract:</strong><br/>
The Bers embebbing realizes the Teichmüller space of a Fuchsian group G as a open,
bounded and contractible subset of the complex Banach space of bounded quadratic differentials for
G. It utilizes the schlicht model of Teichmüller space, where each point is represented by an injective
holomorphic function on the disc, and the map is constructed via the Schwarzian differential operator.
In this paper we prove that a certain class of differential operators acting on functions of the disc
induce holomorphic mappings of Teichmüller spaces, and we also obtain a general formula for the
differential of the induced mappings at the origin. The main focus of this work, however, is on two
particular series of such mappings, dubbed higher Bers maps, because they are induced by so-called
higher Schwarzians – generalizations of the classical Schwarzian operator. For these maps, we prove
several further results.
The last section contains a discussion of possible applications, open questions and speculations.
</p>projecteuclid.org/euclid.ajm/1331663453_Tue, 13 Mar 2012 14:30 EDTTue, 13 Mar 2012 14:30 EDTDeformation of canonical metrics Ihttp://projecteuclid.org/euclid.ajm/1331663454<strong>Xiaofeng Sun</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 1, 141--156.</p>projecteuclid.org/euclid.ajm/1331663454_Tue, 13 Mar 2012 14:30 EDTTue, 13 Mar 2012 14:30 EDTRemarks on the Cartan formula and its applicationshttp://projecteuclid.org/euclid.ajm/1331663455<strong>Kefeng Liu</strong>, <strong>Sheng Rao</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 1, 157--170.</p><p><strong>Abstract:</strong><br/>
In this short note, we present certain generalized versions of the commutator formulas
of some natural operators on manifolds, and give some applications.
</p>projecteuclid.org/euclid.ajm/1331663455_Tue, 13 Mar 2012 14:30 EDTTue, 13 Mar 2012 14:30 EDTOn two-variable primitive $p$-adic $L$-functionshttp://projecteuclid.org/euclid.ajm/1333976881<strong>Yunling Kang</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 171--188.</p><p><strong>Abstract:</strong><br/>
We construct a two variable $p$-adic $L$-function which lead to the $p$-adic interpolation
of values of primitive Hecke $L$-functions, and use it to give a modification of Yager’s theorem which
relate the $p$-adic $L$-function to a certain Iwasawa module.
</p>projecteuclid.org/euclid.ajm/1333976881_Mon, 09 Apr 2012 09:08 EDTMon, 09 Apr 2012 09:08 EDTTotally quasi-umbilic timelike surfaces in $\mathbb{R}^{1,2}$http://projecteuclid.org/euclid.ajm/1333976882<strong>Jeanne Clelland</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 189--208.</p><p><strong>Abstract:</strong><br/>
For a regular surface in Euclidean space $\mathbb{R}^3$, umbilic points are precisely the points
where the Gauss and mean curvatures $K$ and $H$ satisfy $H^2 = K$; moreover, it is well-known that the
only totally umbilic surfaces in $\mathbb{R}^3$ are planes and spheres. But for timelike surfaces in Minkowski
space $\mathbb{R}^{1,2}$, it is possible to have $H^2 = K$ at a non-umbilic point; we call such points quasi-umbilic ,
and we give a complete classification of totally quasi-umbilic timelike surfaces in $\mathbb{R}^{1,2}$.
</p>projecteuclid.org/euclid.ajm/1333976882_Mon, 09 Apr 2012 09:08 EDTMon, 09 Apr 2012 09:08 EDTGeometric flows with rough initial datahttp://projecteuclid.org/euclid.ajm/1333976883<strong>Herbert Koch</strong>, <strong>Tobias Lamm</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 209--235.</p><p><strong>Abstract:</strong><br/>
We show the existence of a global unique and analytic solution for the mean curvature
flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with
small Lipschitz norm. We also show the existence of a global unique and analytic solution to the
Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean
metric in $L^\infty$ and to the harmonic map flow for initial maps whose image is contained in a small
geodesic ball.
</p>projecteuclid.org/euclid.ajm/1333976883_Mon, 09 Apr 2012 09:08 EDTMon, 09 Apr 2012 09:08 EDTOn the normal bundles of rational curves on Fano 3-foldshttp://projecteuclid.org/euclid.ajm/1333976884<strong>Mingmin Shen</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 237--270.</p><p><strong>Abstract:</strong><br/>
A component of very free rational curves on a variety is called unbalanced if the
normal bundle of a general member is unbalanced. In this paper we show that all components
of very free rational curves on a Fano threefold of Picard number one are balanced with the only
exception being the space of conics on $\mathbb{P}^3$.
</p>projecteuclid.org/euclid.ajm/1333976884_Mon, 09 Apr 2012 09:08 EDTMon, 09 Apr 2012 09:08 EDTOn the conjecture of Kosinowskihttp://projecteuclid.org/euclid.ajm/1333976885<strong>Hyun Woong Cho</strong>, <strong>Jin Hong Kim</strong>, <strong>Han Chul Park</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 271--278.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to address some results closely related to the conjecture of
Kosniowski about the number of fixed points on a unitary $S^1$-manifold with only isolated fixed points.
More precisely, if certain $S^1$-equivariant Chern characteristic number of a unitary $S^1$-manifold $M$ is
non-zero, we give a sharp (in certan cases) lower bound on the number of isolated fixed points in
terms of certain integer powers in the $S^1$-equivariant Chern number. In addition, we also deal with
the case of oriented unitary $T^n$-manifolds.
</p>projecteuclid.org/euclid.ajm/1333976885_Mon, 09 Apr 2012 09:08 EDTMon, 09 Apr 2012 09:08 EDTOn the Yau cycle of a normal surface singularityhttp://projecteuclid.org/euclid.ajm/1333976886<strong>Kazuhiro Konno</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 279--298.</p><p><strong>Abstract:</strong><br/>
The notion of the Yau sequence was introduced by Tomaru, as an attempt to extend
Yau’s elliptic sequence for (weakly) elliptic singularities to normal surface singularities of higher
fundamental genera. We show some fundamental properties of the sequence. Among other things,
it is shown that its length gives us the arithmetic genus for singular points of fundamental genus
two. Furthermore, an upper bound on the geometric genus is given for certain surface singularities
of degree one. The relation between the canonical cycle and the Yau cycle is also discussed.
</p>projecteuclid.org/euclid.ajm/1333976886_Mon, 09 Apr 2012 09:08 EDTMon, 09 Apr 2012 09:08 EDTLuttinger surgery and Kodaira dimensionhttp://projecteuclid.org/euclid.ajm/1333976887<strong>Chung-I Ho</strong>, <strong>Tian-Jun Li</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 299--318.</p><p><strong>Abstract:</strong><br/>
In this note we show that the Lagrangian Luttinger surgery preserves the symplectic
Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with nonpositive
Kodaira dimension are also derived.
</p>projecteuclid.org/euclid.ajm/1333976887_Mon, 09 Apr 2012 09:08 EDTMon, 09 Apr 2012 09:08 EDTWeighted thermodynamic formalism on subshifts and applicationshttp://projecteuclid.org/euclid.ajm/1333976888<strong>Julien Barral</strong>, <strong>De-Jun Feng</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 319--352.</p><p><strong>Abstract:</strong><br/>
We examine the interplay between the thermodynamic formalism and the multifractal
formalism on the so-called self-affine symbolic spaces, under the specification property assumption.
We investigate the properties of a weighted variational principle to derive a new result concerning the
approximation of any invariant probability measure $\mu$ by sequences of weighted equilibrium states
whose weighted entropies converge to the weighted entropy of $\mu$. This is a key property in the
estimation of the Hausdorff dimension of sets of generic points, and then in the multifractal analysis
of non homogeneous Birkhoff averages.
</p>projecteuclid.org/euclid.ajm/1333976888_Mon, 09 Apr 2012 09:08 EDTMon, 09 Apr 2012 09:08 EDTFine Selmer group of Hida deformations over non-commutative $p$-adic Lie extensionshttp://projecteuclid.org/euclid.ajm/1333976889<strong>Somnath Jha</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 353--366.</p><p><strong>Abstract:</strong><br/>
We study the Selmer group and the fine Selmer group of $p$-adic Galois representations
defined over a non-commutative $p$-adic Lie extension and their Hida deformations. For the fine Selmer
group, we generalize the pseudonullity conjecture of J. Coates and R. Sujatha, "Fine Selmer group of elliptic curves over $p$-adic Lie extensions," in this context and discuss its invariance
in a branch of a Hida family. We relate the structure of the ‘big’ Selmer (resp. fine Selmer) group
with the specialized individual Selmer (resp. fine Selmer) groups.
</p>projecteuclid.org/euclid.ajm/1333976889_Mon, 09 Apr 2012 09:08 EDTMon, 09 Apr 2012 09:08 EDTOn the Thom-Boardman Symbols for Polynomial Multiplication Mapshttp://projecteuclid.org/euclid.ajm/1353696012<strong>Jiayuan Lin</strong>, <strong>Janice Wethington</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 367--386.</p><p><strong>Abstract:</strong><br/>
The Thom-Boardman symbol was first introduced by Thom in 1956 to classify
singularities of differentiable maps. It was later generalized by Boardman to a more general setting.
Although the Thom-Boardman symbol is realized by a sequence of non-increasing, nonnegative
integers, to compute those numbers is, in general, extremely difficult. In the case of polynomial
multiplication maps, Robert Varley conjectured that computing the Thom-Boardman symbol for
polynomial multiplication reduces to computing the successive quotients and remainders for the Euclidean
algorithm applied to the degrees of the two polynomials. In this paper, we confirm Varley’s
conjecture.
</p>projecteuclid.org/euclid.ajm/1353696012_Fri, 23 Nov 2012 13:40 ESTFri, 23 Nov 2012 13:40 ESTEssentially Large Divisors and their Arithmetic and Function-theoretic Inequalitieshttp://projecteuclid.org/euclid.ajm/1353696013<strong>Gordon Heier</strong>, <strong>Min Ru</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 387--408.</p><p><strong>Abstract:</strong><br/>
Motivated by the classical Theorems of Picard and Siegel and their generalizations,
we define the notion of an essentially large effective divisor and derive some of its arithmetic and
function-theoretic consequences. We then investigate necessary and sufficient criteria for divisors
to be essentially large. In essence, we prove that on a nonsingular irreducible projective variety $X$
with $\mathrm{Pic}(X) = \mathbb{Z}$, every effective divisor with $\operatorname{dim}X + 2$ or more components in general position is
essentially large.
</p>projecteuclid.org/euclid.ajm/1353696013_Fri, 23 Nov 2012 13:40 ESTFri, 23 Nov 2012 13:40 ESTSimple Elliptic Singularities: A Note on their G-functionhttp://projecteuclid.org/euclid.ajm/1353696014<strong>Ian A. B. Strachan</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 409--426.</p><p><strong>Abstract:</strong><br/>
The link between Frobenius manifolds and singularity theory is well known, with
the simplest examples coming from the simple hypersurface singularities. Associated with any such
manifold is a function known as the $G$-function. This plays a role in the construction of higher-genus
terms in various theories. For the simple singularities the $G$-function is known explicitly: $G = 0$ . The
next class of singularities, the unimodal hypersurface or elliptic hypersurface singularities consists of
three examples, $\tilde{E}_6 , \tilde{E}_7 , \tilde{E}_8$ (or equivalently $P_8 ,X_9 , J_10$). Using a result of Noumi and Yamada on
the flat structure on the space of versal deformations of these singularities the $G$-function is explicitly
constructed for these three examples. The main property is that the function depends on only one
variable, the marginal (dimensionless) deformation variable. Other examples are given based on the
foldings of known Frobenius manifolds. Properties of the $G$-function under the action of the modular
group is studied, and applications within the theory of integrable systems are discussed.
</p>projecteuclid.org/euclid.ajm/1353696014_Fri, 23 Nov 2012 13:40 ESTFri, 23 Nov 2012 13:40 ESTErratum to the paper "On Asymptotic Weil-Petersson Geometry of Teichmuller Space of Riemann Surfaces, Asian J. Math., vol. 11, no. 3, 459-484 (2007)"http://projecteuclid.org/euclid.ajm/1353696015<strong>Zheng Huang</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 427--428.</p>projecteuclid.org/euclid.ajm/1353696015_Fri, 23 Nov 2012 13:40 ESTFri, 23 Nov 2012 13:40 ESTNew Thoughts on Weinberger's First and Second Integral Bounds for Green's Functionshttp://projecteuclid.org/euclid.ajm/1353696016<strong>Jie Xiao</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 429--450.</p><p><strong>Abstract:</strong><br/>
New thoughts about the first and second integral bounds of Hans F. Weinberger
for Green’s functions of uniformly elliptic equations are presented by extending the bounds to two
optimal monotone principles, but also further explored via: (i) discovering two new sharp Green-function-
involved isoperimetric inequalities; (ii) verifying the lower dimensional Pólya conjecture for
the lowest eigenvalue of the Laplacian; (iii) sharpening an eccentricity-based lower bound for the
Mahler volumes of the origin-symmetric convex bodies.
</p>projecteuclid.org/euclid.ajm/1353696016_Fri, 23 Nov 2012 13:40 ESTFri, 23 Nov 2012 13:40 ESTDeterminant Line Bundles on Moduli Spaces of Pure Sheaves on Rational Surfaces and Strange Dualityhttp://projecteuclid.org/euclid.ajm/1353696017<strong>Yao Yuan</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 451--478.</p><p><strong>Abstract:</strong><br/>
Let $M^H_X (u)$ be the moduli space of semi-stable pure sheaves of class $u$ on a smooth
complex projective surface $X$. We specify $u = (0, L, \xi(u) = 0)$, i.e. sheaves in $u$ are of dimension 1.
There is a natural morphism $\pi$ from the moduli space $M^H_X (u)$ to the linear system $|L|$. We study a
series of determinant line bundles $\lambda_{c_n^r}$ on $M^H_X (u)$ via $\pi$. Denote $g_L$ the arithmetic genus of curves
in $|L|$. For any $X$ and $g_L \le 0$, we compute the generating function $Z^r(t) = \sum_n h^0(M^H_X (u), \lambda_{c_n^r})t^n$.
For $X$ being $\mathbb{P}^2$ or $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1} (−e))$ with $e = 0, 1$, we compute $Z^1(t)$ for $g_L \gt 0$ and $Z^r(t)$ for all $r$
and $g_L = 1, 2$. Our results provide a numerical check to Strange Duality in these specified situations,
together with Göttsche’s computation. And in addition, we get an interesting corollary (Corollary
4.2.13) in the theory of compactified Jacobian of integral curves.
</p>projecteuclid.org/euclid.ajm/1353696017_Fri, 23 Nov 2012 13:40 ESTFri, 23 Nov 2012 13:40 ESTKähler Immersions of Homogeneous Kähler Manifolds into Complex Space Formshttp://projecteuclid.org/euclid.ajm/1353696018<strong>Antonio Jose Di Scala</strong>, <strong>Hideyuki Ishi</strong>, <strong>Andrea Loi</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 479--488.</p><p><strong>Abstract:</strong><br/>
In this paper we study the homogeneous Kähler manifolds (h.K.m.) which can
be Kähler immersed into finite or infinite dimensional complex space forms. On the one hand we
completely classify the h.K.m. which can be Kähler immersed into a finite or infinite dimensional
complex Euclidean or hyperbolic space. On the other hand, we extend known results about Kähler
immersions into the finite dimensional complex projective space to the infinite dimensional setting.
</p>projecteuclid.org/euclid.ajm/1353696018_Fri, 23 Nov 2012 13:40 ESTFri, 23 Nov 2012 13:40 ESTOn the Structure of the Fundamental Series of Generalized Harish-Chandra Moduleshttp://projecteuclid.org/euclid.ajm/1353696019<strong>Ivan Penkov</strong>, <strong>Gregg Zuckerman</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 489--514.</p><p><strong>Abstract:</strong><br/>
We continue the study of the fundamental series of generalized Harish-Chandra
modules initiated in Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type . Generalized Harish-Chandra modules are $(\mathfrak{g}, \mathfrak{k})$-modules of finite type
where $\mathfrak{g}$ is a semisimple Lie algebra and $\mathfrak{k} \subset \mathfrak{g}$ is a reductive in $\mathfrak{g}$ subalgebra. A first result of the
present paper is that a fundamental series module is a $\mathfrak{g}$-module of finite length. We then define the
notions of strongly and weakly reconstructible simple $(\mathfrak{g}, \mathfrak{k})$-modules $M$ which reflect to what extent
$M$ can be determined via its appearance in the socle of a fundamental series module.
In the second part of the paper we concentrate on the case $\mathfrak{k} \simeq sl(2)$ and prove a sufficient
condition for strong reconstructibility. This strengthens our main result from Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type for the case
$\mathfrak{k} = sl(2)$. We also compute the $sl(2)$-characters of all simple strongly reconstructible (and some
weakly reconstructible) $(\mathfrak{g}, sl(2))$-modules. We conclude the paper by discussing a functor between
a generalization of the category $\mathcal{O}$ and a category of $(\mathfrak{g}, sl(2))$-modules, and we conjecture that this
functor is an equivalence of categories.
</p>projecteuclid.org/euclid.ajm/1353696019_Fri, 23 Nov 2012 13:40 ESTFri, 23 Nov 2012 13:40 ESTFano Threefolds of Genus 6http://projecteuclid.org/euclid.ajm/1353696020<strong>Dmitry Logachev</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 2, 515--560.</p><p><strong>Abstract:</strong><br/>
Ideas and methods of Clemens C. H., Griffiths Ph. The intermediate Jacobian of
a cubic threefold are applied to a Fano threefold $X$ of genus 6 — intersection of $G(2, 5) \subset P^9$ with
$P^7$ and a quadric. Main results:
1. The Fano surface $F(X)$ of $X$ is smooth and irreducible. Hodge numbers and some other
invariants of $F(X)$ are calculated.
2. Tangent bundle theorem for $X$ is proved, and its geometric interpretation is given. It is
shown that $F(X)$ defines $X$ uniquely.
3. The Abel-Jacobi map $\Phi : \operatorname{Alb} F(X) \to J^3(X)$ is an isogeny.
4. As a necessary step of calculation of $h^{1,0}(F(X))$ we describe a special intersection of 3
quadrics in $P^6$ (having 1 double point) whose Hesse curve is a smooth plane curve of degree 6.
5. $\operatorname{im} \Phi(F(X)) \subset J^3(X)$ is algebraically equivalent to $\frac{2\Theta^8}{8!}$
where $\Theta \subset J^3(X)$ is a Poincaré divisor (a sketch of the proof).
</p>projecteuclid.org/euclid.ajm/1353696020_Fri, 23 Nov 2012 13:40 ESTFri, 23 Nov 2012 13:40 ESTFubini-Griffiths-Harris rigidity and Lie algebra cohomologyhttp://projecteuclid.org/euclid.ajm/1355321979<strong>Joseph M. Landsberg</strong>, <strong>Colleen Robles</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 4, 561--586.</p><p><strong>Abstract:</strong><br/>
We prove a rigidity theorem for represented semi-simple Lie groups. The theorem is
used to show that the adjoint variety of a complex simple Lie algebra $\mathfrak{g}$ (the unique minimal $G$ orbit
in $\mathbb{P}_{\mathfrak{g}}$) is extrinsically rigid to third order (with the exception of $\mathfrak{g} = \mathfrak{a}_1$).
In contrast, we show that the adjoint variety of $SL_3\mathbb{C}$ and the Segre product $Seg(\mathbb{P}^1 \times \mathbb{P}^n$)
are flexible at order two. In the $SL_3\mathbb{C}$ example we discuss the relationship between the extrinsic
projective geometry and the intrinsic path geometry.
We extend machinery developed by Hwang and Yamaguchi, Se-ashi, Tanaka and others to reduce
the proof of the general theorem to a Lie algebra cohomology calculation. The proofs of the flexibility
statements use exterior differential systems techniques.
</p>projecteuclid.org/euclid.ajm/1355321979_Wed, 12 Dec 2012 09:19 ESTWed, 12 Dec 2012 09:19 ESTOn a construction of Burago and Zalgallerhttp://projecteuclid.org/euclid.ajm/1355321980<strong>Emil Saucan</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 4, 587--606.</p><p><strong>Abstract:</strong><br/>
The purpose of this note is to scrutinize the proof of Burago and Zalgaller regarding
the existence of $PL$ isometric embeddings of $PL$ compact surfaces into $\mathbb{R}^3$. We conclude that their
proof does not admit a direct extension to higher dimensions. Moreover, we show that, in general,
$PL$ manifolds of dimension $n \ge 3$ admit no nontrivial $PL$ embeddings in $\mathbb{R}^{n+1}$ that are close to
conformality. We also extend the result of Burago and Zalgaller to a large class of noncompact
$PL$ 2-manifolds. The relation between intrinsic and extrinsic curvatures is also examined, and we
propose a $PL$ version of the Gauss compatibility equation for smooth surfaces.
</p>projecteuclid.org/euclid.ajm/1355321980_Wed, 12 Dec 2012 09:19 ESTWed, 12 Dec 2012 09:19 ESTKac-Moody groups, infinite dimensional differential geometry and citieshttp://projecteuclid.org/euclid.ajm/1355321981<strong>Walter Freyn</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 4, 607--636.</p><p><strong>Abstract:</strong><br/>
Minimal affine Kac-Moody groups act on affine twin buildings by isometries. However
there is no way to extend this action to any completion of the Kac-Moody groups. To remedy
this, we introduce in this paper affine twin cities, a new class of objects, whose elements behave
like completions of twin buildings. Twin cities are defined as special arrays of affine buildings
connected among themselves by twinnings. Corresponding to completed affine Kac-Moody groups
they are characterized by the type of the affine buildings and by some kind of "regularity conditions"
describing the completion. The isometry groups of affine twin cities are (completions of) affine
Kac-Moody groups. We study applications of cities in infinite dimensional differential geometry by
proving infinite dimensional versions of classical differential geometric results: For example, we show
that points in an isoparametric submanifold in a Hilbert space correspond to all chambers in a city.
In two sequels we will describe the theory of twin cities for formal completions.
</p>projecteuclid.org/euclid.ajm/1355321981_Wed, 12 Dec 2012 09:19 ESTWed, 12 Dec 2012 09:19 ESTHarmonic forms on principal bundleshttp://projecteuclid.org/euclid.ajm/1355321982<strong>Corbett Redden</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 4, 637--660.</p><p><strong>Abstract:</strong><br/>
We show a relationship between Chern–Simons 1- and 3-forms and harmonic forms
on a principal bundle. Doing so requires one to consider an adiabatic limit. For the 3-form case,
assume that $G$ is simple and the corresponding Chern–Weil 4-form is exact. Then, the Chern–Simons
3-form on the princpal bundle $G$-bundle, minus a canonical term from the base, is harmonic in the
adiabatic limit.
</p>projecteuclid.org/euclid.ajm/1355321982_Wed, 12 Dec 2012 09:19 ESTWed, 12 Dec 2012 09:19 ESTMorse field theoryhttp://projecteuclid.org/euclid.ajm/1355321983<strong>Ralph Cohen</strong>, <strong>Paul Norbury</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 4, 661--712.</p><p><strong>Abstract:</strong><br/>
In this paper we define and study the moduli space of metric-graph-flows in a manifold
$M$. This is a space of smooth maps from a finite graph to $M$, which, when restricted to each edge, is
a gradient flow line of a smooth (and generically Morse) function on $M$. Using the model of Gromov-
Witten theory, with this moduli space replacing the space of stable holomorphic curves in a symplectic
manifold, we obtain invariants, which are (co)homology operations in $M$. The invariants obtained in
this setting are classical cohomology operations such as cup product, Steenrod squares, and Stiefel-
Whitney classes. We show that these operations satisfy invariance and gluing properties that fit
together to give the structure of a topological quantum field theory. By considering equivariant
operations with respect to the action of the automorphism group of the graph, the field theory has
more structure. It is analogous to a homological conformal field theory. In particular we show that
classical relations such as the Adem relations and Cartan formulae are consequences of these field
theoretic properties. These operations are defined and studied using two different methods. First,
we use algebraic topological techniques to define appropriate virtual fundamental classes of these
moduli spaces. This allows us to define the operations via the corresponding intersection numbers of
the moduli space. Secondly, we use geometric and analytic techniques to study the smoothness and
compactness properties of these moduli spaces. This will allow us to define these operations on the
level of Morse-Smale chain complexes, by appropriately counting metric-graph-flows with particular
boundary conditions.
</p>projecteuclid.org/euclid.ajm/1355321983_Wed, 12 Dec 2012 09:19 ESTWed, 12 Dec 2012 09:19 ESTDeformations of nearly parallel $G_2$-structureshttp://projecteuclid.org/euclid.ajm/1355321984<strong>Bogdan Alexandrov</strong>, <strong>Uwe Semmelmann</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 4, 713--744.</p><p><strong>Abstract:</strong><br/>
We study the infinitesimal deformations of a proper nearly parallel $G_2$-structure and
prove that they are characterized by a certain first order differential equation. In particular we show
that the space of infinitesimal deformations modulo the group of diffeomorphisms is isomorphic to a
subspace of co-closed $\Lambda^3_{27}$-eigenforms of the Laplace operator for the eigenvalue $8\mathrm{scal} /21$. We give a
similar description for the space of infinitesimal Einstein deformations of a fixed nearly parallel $G_2$-structure. Moreover we show
that there are no deformations on the squashed $S^7$ and on $\mathrm{SO}(5)/\mathrm{SO}(3)$,
but that there are infinitesimal deformations on the Aloff-Wallach manifold $N(1, 1) = \mathrm{SU}(3)/U(1)$.
</p>projecteuclid.org/euclid.ajm/1355321984_Wed, 12 Dec 2012 09:19 ESTWed, 12 Dec 2012 09:19 ESTCalabi-Yau maniolds and generic Hodge groupshttp://projecteuclid.org/euclid.ajm/1355321985<strong>Jan Christian Rohde</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 4, 745--774.</p><p><strong>Abstract:</strong><br/>
We study the generic Hodge groups $\mathrm{Hg}(\mathcal{X})$ of local universal deformations $\mathcal{X}$ of
Calabi-Yau 3-manifolds with onedimensional complex moduli, give a complete list of all possible
choices for $\mathrm{Hg}(\mathcal{X})_{\mathbb{R}}$ and determine the latter real groups for known examples.
</p>projecteuclid.org/euclid.ajm/1355321985_Wed, 12 Dec 2012 09:19 ESTWed, 12 Dec 2012 09:19 ESTScaling of Poisson spheres and compact Lie groupshttp://projecteuclid.org/euclid.ajm/1355321986<strong>Albert Jeu-Liang Sheu</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 4, 775--786.</p><p><strong>Abstract:</strong><br/>
For $n \ge 2$, we show that on the standard Poisson homogeneous space $\mathbb{S}^{2n−1}$ (including
$SU (2) \approx \mathbb{S}3$), there exists a Poisson scaling $\phi_\lambda$ at any scale $\lambda \gt 0$ that is smooth on each
symplectic leaf and continuous globally. A generalization to the case of the standard Bruhat-Poisson
compact simple Lie groups endowed with a stronger topology is also valid.
</p>projecteuclid.org/euclid.ajm/1355321986_Wed, 12 Dec 2012 09:19 ESTWed, 12 Dec 2012 09:19 ESTThe class of a Hurwitz divisor on the moduli of curves of even genushttp://projecteuclid.org/euclid.ajm/1355321987<strong>Gerard van der Geer</strong>, <strong>Alexis Kouvidakis</strong><p><strong>Source: </strong>Asian J. Math., Volume 16, Number 4, 787--806.</p><p><strong>Abstract:</strong><br/>
We study the geometry of the natural map from the Hurwitz space $\overline{H}_{2k,k+1}$ to the
moduli space $\overline{\mathcal{M}}_{2k}$. We calculate the cycle class of the Hurwitz divisor $D_2$ on $\overline{\mathcal{M}}_g$ for $g = 2k$ given
by the degree $k + 1$ covers of $\mathbb{P}^1$ with simple ramification points, two of which lie in the same fibre.
This has applications to bounds on the slope of the moving cone of $\overline{\mathcal{M}}_g$, the calculation of other
divisor classes and motivated an algebraic proof for the formula of the Hodge bundle of the Hurwitz
space.
</p>projecteuclid.org/euclid.ajm/1355321987_Wed, 12 Dec 2012 09:19 ESTWed, 12 Dec 2012 09:19 ESTOn the existence of pseudoharmonic maps from pseudohermitian manifolds into Riemannian manifolds with nonpositive sectional curvaturehttp://projecteuclid.org/euclid.ajm/1383923433<strong>Shu-Cheng Chang</strong>, <strong>Ting-Hui Chang</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 1, 1--16.</p><p><strong>Abstract:</strong><br/>
In this paper, we first derive a CR Bochner identity for the pseudoharmonic map heat
flow on pseudohermitian manifolds. Secondly, we are able to prove existence of the global solution
for the pseudoharmonic map heat flow from a closed pseudohermitian manifold into a Riemannian
manifold with nonpositive sectional curvature. In particular, we prove the existence theorem of
pseudoharmonic maps. This is served as the CR analogue of Eells-Sampson’s Theorem for the
harmonic map heat flow.
</p>projecteuclid.org/euclid.ajm/1383923433_Fri, 08 Nov 2013 10:10 ESTFri, 08 Nov 2013 10:10 ESTLower diameter bounds for compact shrinking Ricci solitonshttp://projecteuclid.org/euclid.ajm/1383923434<strong>Akito Futaki</strong>, <strong>Yuji Sano</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 1, 17--32.</p><p><strong>Abstract:</strong><br/>
It is shown that the diameter of a compact shrinking Ricci soliton has a universal
lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of
Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian
on compact shrinking Ricci solitons.
</p>projecteuclid.org/euclid.ajm/1383923434_Fri, 08 Nov 2013 10:10 ESTFri, 08 Nov 2013 10:10 ESTCohomogeneity one shrinking Ricci solitons: An analytic and numerical studyhttp://projecteuclid.org/euclid.ajm/1383923435<strong>Andrew S. Dancer</strong>, <strong>Stuart J. Hall</strong>, <strong>McKenzie Y. Wang</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 1, 33--62.</p><p><strong>Abstract:</strong><br/>
We use analytical and numerical methods to investigate the equations for cohomogeneity
one shrinking gradient Ricci solitons. We show the existence of a winding number for this
system around the subvariety of phase space corresponding to Einstein solutions and obtain some
estimates for it. We prove a non-existence result for certain orbit types, analogous to that of Böhm
in the Einstein case. We also carry out numerical investigations for selected orbit types.
</p>projecteuclid.org/euclid.ajm/1383923435_Fri, 08 Nov 2013 10:10 ESTFri, 08 Nov 2013 10:10 ESTA wall crossing formula of Donaldson-Thomas invariants without Chern-Simons functionalhttp://projecteuclid.org/euclid.ajm/1383923436<strong>Young-Hoon Kiem</strong>, <strong>Jun Li</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 1, 63--94.</p><p><strong>Abstract:</strong><br/>
We prove a wall crossing formula of Donaldson-Thomas type invariants without Chern-Simons functionals.
</p>projecteuclid.org/euclid.ajm/1383923436_Fri, 08 Nov 2013 10:10 ESTFri, 08 Nov 2013 10:10 ESTEverywhere equivalent and everywhere different knot diagramshttp://projecteuclid.org/euclid.ajm/1383923437<strong>Alexander Stoimenow</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 1, 95--138.</p><p><strong>Abstract:</strong><br/>
A knot diagram is said to be everywhere different (resp. everywhere equivalent) if
all the diagrams obtained by switching one crossing represent different (resp. the same) knot(s). We
exhibit infinitely many everywhere different knot diagrams. We also present several constructions
of everywhere equivalent knot diagrams, and prove that among certain classes these constructions
are exhaustive. Finally, we consider a generalization to link diagrams, and discuss some relation to
symmetry properties of planar graphs.
</p>projecteuclid.org/euclid.ajm/1383923437_Fri, 08 Nov 2013 10:10 ESTFri, 08 Nov 2013 10:10 ESTYang-Mills connections of cohomogeneity one on SO(n)-bundles over Euclidean sphereshttp://projecteuclid.org/euclid.ajm/1383923438<strong>Andreas Gastel</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 1, 139--162.</p>projecteuclid.org/euclid.ajm/1383923438_Fri, 08 Nov 2013 10:10 ESTFri, 08 Nov 2013 10:10 ESTRigid flat web on the projective planehttp://projecteuclid.org/euclid.ajm/1383923439<strong>David Marín</strong>, <strong>Jorge Vitório Pereira</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 1, 163--192.</p><p><strong>Abstract:</strong><br/>
This paper studies global webs on the projective plane with vanishing curvature. The
study is based on an interplay of local and global arguments. The main local ingredient is a criterium
for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The
main global ingredient, the Legendre transform, is an avatar of classical projective duality in the
realm of differential equations. We show that the Legendre transform of what we call reduced convex
foliations are webs with zero curvature, and we exhibit a countable infinity family of convex foliations
which give rise to a family of webs with zero curvature not admitting non-trivial deformations with
zero curvature.
</p>projecteuclid.org/euclid.ajm/1383923439_Fri, 08 Nov 2013 10:10 ESTFri, 08 Nov 2013 10:10 ESTTautological module and intersection theory on Hilbert schemes of nodal curveshttp://projecteuclid.org/euclid.ajm/1383923851<strong>Ziv Ran</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 2, 193--264.</p><p><strong>Abstract:</strong><br/>
This paper presents the rudiments of Hilbert-Mumford Intersection (HMI) theory:
intersection theory on the relative Hilbert scheme of a family of nodal (or smooth) curves, over a
base of arbitrary dimension. We introduce an additive group of geometric cycles, called ’tautological
module’, generated by diagonal loci, node scrolls, and twists thereof. We determine recursively the
intersection action on this group by the discriminant (big diagonal) divisor and all its powers. We
show that this suffices to determine arbitrary polynomials in Chern classes, in particular Chern
numbers, for the tautological vector bundles on the Hilbert schemes, which are closely related to
enumerative geometry of families of nodal curves.
</p>projecteuclid.org/euclid.ajm/1383923851_Fri, 08 Nov 2013 10:17 ESTFri, 08 Nov 2013 10:17 ESTThe Atiyah-Patodi-Singer index theorem for Dirac operators over C*-algebrashttp://projecteuclid.org/euclid.ajm/1383923852<strong>Charlotte Wahl</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 2, 265--320.</p><p><strong>Abstract:</strong><br/>
We prove a higher Atiyah–Patodi–Singer index theorem for Dirac operators twisted by $C^*$-vector bundles. We use it to derive a general product formula for $\eta$-forms and to define and study
new $\rho$-invariants generalizing Lott’s higher $\rho$-form. The higher Atiyah–Patodi–Singer index theorem of Leichtnam–Piazza can be recovered by applying the theorem to Dirac operators
twisted by the Mishenko–Fomenko bundle associated to the reduced $C^*$-algebra of the fundamental group.
</p>projecteuclid.org/euclid.ajm/1383923852_Fri, 08 Nov 2013 10:17 ESTFri, 08 Nov 2013 10:17 ESTExistence of compatible contact structures on $G_2$-manifoldshttp://projecteuclid.org/euclid.ajm/1383923853<strong>M. Firat Arikan</strong>, <strong>Hyunjoo Cho</strong>, <strong>Sema Salur</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 2, 321--334.</p><p><strong>Abstract:</strong><br/>
In this paper, we show the existence of (co-oriented) contact structures on certain classes of $G_2$-manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any
seven-manifold with a spin structure (and so any manifold with $G_2$-structure) admits an almost contact structure. We also construct explicit almost contact metric structures on manifolds with $G_2$-structures.
</p>projecteuclid.org/euclid.ajm/1383923853_Fri, 08 Nov 2013 10:17 ESTFri, 08 Nov 2013 10:17 ESTArithmetic intersection on a Hilbert modular surface and the Faltings heighthttp://projecteuclid.org/euclid.ajm/1383923854<strong>Tonghai Yang</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 2, 335--382.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove an explicit arithmetic intersection formula between arithmetic Hirzebruch-Zagier divisors and arithmetic CM cycles on a Hilbert modular surface over $\mathbb{Z}$. As applications, we obtain
the first ‘non-abelian’ Chowla-Selberg formula, which is a special case of Colmez’s conjecture; an explicit arithmetic intersection formula between arithmetic Humbert surfaces and CM cycles in the arithmetic
Siegel modular variety of genus two; Lauter’s conjecture about the denominators of CM values of Igusa invariants; and a result about bad reduction of CM genus two curves.
</p>projecteuclid.org/euclid.ajm/1383923854_Fri, 08 Nov 2013 10:17 ESTFri, 08 Nov 2013 10:17 ESTOn an algebraic formula and applications to group action on manifoldshttp://projecteuclid.org/euclid.ajm/1383923855<strong>Ping Li</strong>, <strong>Kefeng Liu</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 2, 383--390.</p><p><strong>Abstract:</strong><br/>
In this paper we consider a purely algebraic result. Then given a circle or cyclic group of prime order action on a manifold, we will use it to estimate the lower bound of the number of fixed points. We also give an
obstruction to the existence of $\mathbb{Z}_p$ action on manifolds with isolated fixed points when $p$ is a prime.
</p>projecteuclid.org/euclid.ajm/1383923855_Fri, 08 Nov 2013 10:17 ESTFri, 08 Nov 2013 10:17 ESTA combinatorial invariant for spherical CR structureshttp://projecteuclid.org/euclid.ajm/1383923951<strong>Elisha Falbel</strong>, <strong>Qingxue Wang</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 3, 391--422.</p><p><strong>Abstract:</strong><br/>
We study a cross-ratio of four generic points of $S^3$ which comes from spherical CR geometry. We construct a homomorphism from a certain group generated by generic configurations of four points
in $S^3$ to the pre-Bloch group $\mathcal{P}(\mathbb{C})$. If $M$ is a 3-dimensional spherical CR manifold with a CR triangulation, by our homomorphism, we get a $\mathcal{P}(\mathbb{C})$-valued
invariant for $M$. We show that when applying to it the Bloch-Wigner function, it is zero. Under some conditions on $M$, we show the invariant lies in the Bloch group $\mathcal{B}(k)$, where $k$ is the
field generated by the cross-ratio. For a CR triangulation of the Whitehead link complement, we show its invariant is a torsion in $\mathcal{B}(k)$ and for a triangulation of the complement of the 52-knot
we show that the invariant is not trivial and not a torsion element.
</p>projecteuclid.org/euclid.ajm/1383923951_Fri, 08 Nov 2013 10:19 ESTFri, 08 Nov 2013 10:19 ESTMinimality of symplectic fiber sums along sphereshttp://projecteuclid.org/euclid.ajm/1383923952<strong>Josef G. Dorfmeister</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 3, 423--442.</p><p><strong>Abstract:</strong><br/>
In this note we complete the discussion begun in A. I. Stipsicz, Indecomposability of certain Lefschetz fibrations , concerning the minimality of symplectic fiber sums.
We find that for fiber sums along spheres the minimality of the sum is determined by the cases discussed
in M. Usher, Minimality and symplectic sums , and one additional case: If $X{\#}_VY = Z {\#}V_{\mathbb{C}P^2}\mathbb{C}P^2$ with $V_{\mathbb{C}P^2}$
an embedded +4-sphere in class $[V_{\mathbb{C}P^2}] = 2[H] \in H_2(\mathbb{C}P_2, Z)$ and
$Z$ has at least 2 disjoint exceptional spheres $E_i$ each meeting the submanifold $V_Z \subset Z$ positively and transversely in a single point, then the fiber sum is not minimal.
</p>projecteuclid.org/euclid.ajm/1383923952_Fri, 08 Nov 2013 10:19 ESTFri, 08 Nov 2013 10:19 ESTVolume growth eigenvalue and compactness for self-shrinkershttp://projecteuclid.org/euclid.ajm/1383923953<strong>Qi Ding</strong>, <strong>Y. L. Xin</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 3, 443--456.</p><p><strong>Abstract:</strong><br/>
In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue
conjecture on minimal hypersurfaces in the unit sphere by Yau. By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in $\mathbb{R}^3$ obtained
by Colding-Minicozzi under weaker conditions.
</p>projecteuclid.org/euclid.ajm/1383923953_Fri, 08 Nov 2013 10:19 ESTFri, 08 Nov 2013 10:19 ESTRemarks on a scalar curvature rigidity theorem of Brendle and Marqueshttp://projecteuclid.org/euclid.ajm/1383923954<strong>Graham Cox</strong>, <strong>Pengzi Miao</strong>, <strong>Luen-Fai Tam</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 3, 457--470.</p><p><strong>Abstract:</strong><br/>
We give an improvement of a scalar curvature rigidity theorem of Brendle and Marques regarding geodesic balls in $\mathbb{S}^n$. The main result is that Brendle and Marques' theorem holds on a geodesic ball larger
than that specified in Scalar curvature rigidity of geodesic balls in $\mathbb{S}^n$ .
</p>projecteuclid.org/euclid.ajm/1383923954_Fri, 08 Nov 2013 10:19 ESTFri, 08 Nov 2013 10:19 ESTFloer homology for 2-torsion instanton invariantshttp://projecteuclid.org/euclid.ajm/1383923955<strong>Hirofumi Sasahira</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 3, 471--524.</p><p><strong>Abstract:</strong><br/>
We construct a variant of Floer homology groups and prove a gluing formula for a
variant of Donaldson invariants. As a corollary, the variant of Donaldson invariants is non-trivial for
connected sums of 4-manifolds which satisfy a condition for Donaldson invariants. We also show a
non-existence result of compact, spin 4-manifolds with boundary some homology 3-spheres and with
certain intersection forms.
</p>projecteuclid.org/euclid.ajm/1383923955_Fri, 08 Nov 2013 10:19 ESTFri, 08 Nov 2013 10:19 ESTA geometric theory of zero area singularities in general relativityhttp://projecteuclid.org/euclid.ajm/1383923956<strong>Hubert L. Bray</strong>, <strong>Jeffrey L. Jauregui</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 3, 525--560.</p><p><strong>Abstract:</strong><br/>
The Schwarzschild spacetime metric of negative mass is well-known to contain a naked
singularity. In a spacelike slice, this singularity of the metric is characterized by the property that
nearby surfaces have arbitrarily small area. We develop a theory of such "zero area singularities"
in Riemannian manifolds, generalizing far beyond the Schwarzschild case (for example, allowing
the singularities to have nontrivial topology). We also define the mass of such singularities. The
main result of this paper is a lower bound on the ADM mass of an asymptotically
at manifold of nonnegative scalar curvature in terms of the masses of its singularities, assuming a certain conjecture
in conformal geometry. The proof relies on the Riemannian Penrose inequality. Equality is
attained in the inequality by the Schwarzschild metric of negative mass. An immediate corollary is
a version of the positive mass theorem that allows for certain types of incomplete metrics.
</p>projecteuclid.org/euclid.ajm/1383923956_Fri, 08 Nov 2013 10:19 ESTFri, 08 Nov 2013 10:19 ESTManifolds with nef contangent bundlehttp://projecteuclid.org/euclid.ajm/1383923957<strong>Andreas Höring</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 3, 561--568.</p><p><strong>Abstract:</strong><br/>
Generalising a classical theorem by Ueno, we prove structure results for manifolds
with nef or semiample cotangent bundle.
</p>projecteuclid.org/euclid.ajm/1383923957_Fri, 08 Nov 2013 10:19 ESTFri, 08 Nov 2013 10:19 ESTLogarithmic Sobolev trace inequalitieshttp://projecteuclid.org/euclid.ajm/1383923958<strong>Filomena Feo</strong>, <strong>Maria Rosaria Posteraro</strong><p><strong>Source: </strong>Asian J. Math., Volume 17, Number 3, 569--582.</p><p><strong>Abstract:</strong><br/>
We prove a logarithmic Sobolev trace inequality and we study the trace operator in the weighted Sobolev space $W^{1,p} (\Omega , \gamma)$ for sufficiently regular domain, where $\gamma$ is the Gauss measure.
Applications to PDE are also considered.
</p>projecteuclid.org/euclid.ajm/1383923958_Fri, 08 Nov 2013 10:19 ESTFri, 08 Nov 2013 10:19 EST