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 EDTA 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 ESTCharacterizations of projective spaces and hyperquadricshttp://projecteuclid.org/euclid.ajm/1408712343<strong>Stéphane Druel</strong>, <strong>Matthieu Paris</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 17, Number 4, 583--596.</p><p><strong>Abstract:</strong><br/>
In this paper we prove that if the $r$-th tensor power of the tangent bundle of a smooth projective variety $X$ contains the determinant of an ample vector bundle of rank at least $r$, then $X$ is isomorphic either to
a projective space or to a smooth quadric hypersurface. Our result generalizes Mori's, Wahl's, Andreatta-Wiśniewski's and Araujo-Druel-Kovács's characterizations of projective spaces and hyperquadrics.
</p>projecteuclid.org/euclid.ajm/1408712343_20140822085905Fri, 22 Aug 2014 08:59 EDTA topological approach to unifying compactifications of symmetric spaceshttp://projecteuclid.org/euclid.ajm/1408712344<strong>Pedro J. Freitas</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 17, Number 4, 597--608.</p><p><strong>Abstract:</strong><br/>
In this paper we present a topological way of building a compactification of a symmetric space from a compactification of a Weyl Chamber. We also present a way to achieve compactifications by means of generalized
Busemann functions.
</p>projecteuclid.org/euclid.ajm/1408712344_20140822085905Fri, 22 Aug 2014 08:59 EDTAlgebro-geometric semistability of polarized toric manifoldshttp://projecteuclid.org/euclid.ajm/1408712345<strong>Hajime Ono</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 17, Number 4, 609--616.</p><p><strong>Abstract:</strong><br/>
Let $\Delta \subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_{\Delta}$ and a very ample
$(\mathbb{C}×)^n$-equivariant line bundle $L_{\Delta}$ on $X_{\Delta}$ associated with $\Delta$. In the present paper, we give a necessary and sufficient condition for Chow semistability
of $( X_{\Delta}, {L^i}_{\Delta})$ for a maximal torus action. We then see that asymptotic (relative) Chow semistability implies (relative) K-semistability for toric degenerations, which is proved by Ross and Thomas.
</p>projecteuclid.org/euclid.ajm/1408712345_20140822085905Fri, 22 Aug 2014 08:59 EDTThe overconvergent frobeniushttp://projecteuclid.org/euclid.ajm/1408712346<strong>Robert F. Coleman</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 17, Number 4, 617--620.</p>projecteuclid.org/euclid.ajm/1408712346_20140822085905Fri, 22 Aug 2014 08:59 EDTSpacelike foliations by $(n−1)$-umbilical hypersurfaces in spacetimeshttp://projecteuclid.org/euclid.ajm/1408712347<strong>A. Gervasio Colares</strong>, <strong>Oscar Palmas</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 17, Number 4, 621--644.</p><p><strong>Abstract:</strong><br/>
We consider the problem of whether a given spacetime admits a foliation by $(n−1)$-umbilical spacelike hypersurfaces. We introduce the notion of a timelike closed partially conformal vector field in a spacetime and
show that the existence of a vector field of this kind guarantees in turn the existence of that foliation. We then construct explicit examples of families of $(n−1)$-umbilical spacelike hypersurfaces in the de Sitter space.
Imposing the further condition of having constant $r$-th mean curvature, we give the complete description of any leaf of a foliation of the de Sitter space by these hypersurfaces. Finally, in a spacetime foliated
by $(n−1)$-umbilical spacelike hypersurfaces we characterize the immersed spacelike hypersurfaces which are $(n−1)$-umbilical.
</p>projecteuclid.org/euclid.ajm/1408712347_20140822085905Fri, 22 Aug 2014 08:59 EDTHomotopy minimal period self-maps on flat manifolds with cyclic holonomieshttp://projecteuclid.org/euclid.ajm/1408712348<strong>Zhibin Liang</strong>, <strong>Xuezhi Zhao</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 17, Number 4, 645--652.</p><p><strong>Abstract:</strong><br/>
This paper studies the homotopical minimal period of self-maps on flat manifolds with cyclic holonomies. We give some necessary conditions on the self-maps on flat manifolds to guarantee that their homotopical
minimal periods are infinite. Furthermore, a kind of density of homotopical minimal periods in the natural number set is considered.
</p>projecteuclid.org/euclid.ajm/1408712348_20140822085905Fri, 22 Aug 2014 08:59 EDTDynamic equivalence of control systems via infinite prolongationhttp://projecteuclid.org/euclid.ajm/1408712349<strong>Matthew W. Stackpole</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 17, Number 4, 653--688.</p><p><strong>Abstract:</strong><br/>
In this paper, we put the issue of dynamic equivalence of control systems in the context of pullbacks of coframings on infinite jet bundles over the state manifolds. While much attention has been given to differentially
flat systems, i.e., systems dynamically equivalent to linear control systems, the advantage of this approach is that it allowed us to consider control affine systems as well. Through this context we are able to
classify all control affine systems of three states and two controls under dynamic equivalence of the type $(x,u)\mapsto y(x,u)$.
</p>projecteuclid.org/euclid.ajm/1408712349_20140822085905Fri, 22 Aug 2014 08:59 EDTNoether's problem and unramified Brauer groupshttp://projecteuclid.org/euclid.ajm/1408712350<strong>Akinari Hoshi</strong>, <strong>Ming-Chang Kang</strong>, <strong>Boris E. Kunyavskii</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 17, Number 4, 689--714.</p><p><strong>Abstract:</strong><br/>
Let $k$ be any field, $G$ be a finite group acting on the rational function field $k(x_g : g \in G)$ by $h \cdot x_g = x_{hg}$ for any $h, g \in G$. Define $k(G) = k(x_g : g \in G)^G$. Noether's problem asks
whether $k(G)$ is rational (= purely transcendental) over $k$. It is known that, if $\mathbb{C}(G)$ is rational over $\mathbb{C}$, then $B_0(G) = 0$ where $B_0(G)$ is the unramified Brauer group
of $\mathbb{C}(G)$ over $\mathbb{C}$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G) = 0$. This result was disproved by Moravec for $p = 3, 5, 7$ by computer calculations.
We will prove the following theorem. Theorem. Let $p$ be any odd prime number, $G$ be a group of order $p^5$. Then $B_0(G) \neq 0$ if and only if $G$ belongs to the isoclinism family ${\Phi}_{10}$ in
R. James's classification of groups of order $p^5$.
</p>projecteuclid.org/euclid.ajm/1408712350_20140822085905Fri, 22 Aug 2014 08:59 EDTA new proof of almost isometry theorem in Alexandrov geometry with curvature bounded belowhttp://projecteuclid.org/euclid.ajm/1408712351<strong>Yusheng Wang</strong>, <strong>Xiaole Su</strong>, <strong>Hongwei Sun</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 17, Number 4, 715--728.</p><p><strong>Abstract:</strong><br/>
In this paper we give a new proof (along the line of the original proof) for the almost isometry theorem in Alexandrov geometry with curvature bounded below in Yu. Burago, M. Gromov, and G. Perel’man, A. D. Alexandrov
spaces with curvature bounded below , Russian Math. Surveys, 47:2 (1992), pp. 1–58. The motivation of the new proof is that we find that Lemma 9.11 in A. D. Alexandrov
spaces with curvature bounded below is incorrect (see Example 1.3 below), while this lemma is a crucial step in the original proof.
</p>projecteuclid.org/euclid.ajm/1408712351_20140822085905Fri, 22 Aug 2014 08:59 EDTEigenvalues of Hecke operators on Hilbert modular groupshttp://projecteuclid.org/euclid.ajm/1408712352<strong>Roelof W. Bruggeman</strong>, <strong>Roberto J. Miatello</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 17, Number 4, 729--758.</p>projecteuclid.org/euclid.ajm/1408712352_20140822085905Fri, 22 Aug 2014 08:59 EDTGeneralized existence of isoperimetric regions in non-compact Riemannian manifolds and applications to the isoperimetric profilehttp://projecteuclid.org/euclid.ajm/1409168510<strong>Stefano Nardulli</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 1, 1--28.</p><p><strong>Abstract:</strong><br/>
For a complete noncompact Riemannian manifold with smoothly bounded geometry,
we prove the existence of isoperimetric regions in a larger space obtained by adding finitely many limit
manifolds at infinity. As one of many possible applications, we extend properties of the isoperimetric
profile from compact manifolds to such noncompact manifolds.
</p>projecteuclid.org/euclid.ajm/1409168510_20140827154152Wed, 27 Aug 2014 15:41 EDTA remark on Mirzakhani's asymptotic formulaehttp://projecteuclid.org/euclid.ajm/1409168511<strong>Kefeng Liu</strong>, <strong>Hao Xu</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 1, 29--52.</p><p><strong>Abstract:</strong><br/>
We give a short proof of Penner-Grushevsky-Schumacher-Trapani’s large genus asymptotics of Weil-Petersson volumes of moduli spaces of curves. We also study asymptotic expansions for certain integrals
of pure $\psi$ classes and answer a question of Mirzakhani on the asymptotic behavior of one-point volume polynomials of moduli spaces of curves.
</p>projecteuclid.org/euclid.ajm/1409168511_20140827154152Wed, 27 Aug 2014 15:41 EDTIsoparametric hypersurfaces and metrics of constant scalar curvaturehttp://projecteuclid.org/euclid.ajm/1409168512<strong>Guillermo Henry</strong>, <strong>Jimmy Petean</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 1, 53--68.</p><p><strong>Abstract:</strong><br/>
We showed the existence of non-radial solutions of the equation $\Delta u - \lambda u + \lambda u^q = 0$ on the round sphere $S^m$, for $q \lt (m + 2)/ (m - 2)$, and study the number of such solutions in terms
of $\lambda$. We show that for any isoparametric hypersurface $M \subset S^m$ there are solutions such that $M$ is a regular level set (and the number of such solutions increases with $\lambda$). We also show
similar results for isoparametric hypersurfaces in general Riemannian manifolds. These solutions give multiplicity results for metrics of constant scalar curvature on conformal classes of Riemannian products.
</p>projecteuclid.org/euclid.ajm/1409168512_20140827154152Wed, 27 Aug 2014 15:41 EDTKähler manifolds with Ricci curvature lower bondhttp://projecteuclid.org/euclid.ajm/1409168513<strong>Gang Liu</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 1, 69--100.</p><p><strong>Abstract:</strong><br/>
On Kähler manifolds with Ricci curvature bounded from below, we establish some
theorems which are counterparts of some classical theorems in Riemannian geometry, for example,
Bishop-Gromov’s relative volume comparison, Bonnet-Meyers theorem, and Yau’s gradient estimate
for positive harmonic functions. The tool is a Bochner type formula reflecting the Kähler structure.
</p>projecteuclid.org/euclid.ajm/1409168513_20140827154152Wed, 27 Aug 2014 15:41 EDTFirst order deformations of pairs of a rational curve and a hypersurfacehttp://projecteuclid.org/euclid.ajm/1409168514<strong>Bin Wang</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 1, 101--116.</p><p><strong>Abstract:</strong><br/>
Let $X_0$ be a smooth hypersurface (not assumed generic) in projective space $\mathrm{P}^n$, $n \geq 3$ over the complex numbers, and $C_0$ a smooth rational curve on $X_0$. We are interested in the
deformations of the pair $C_0 , X_0$. In this paper, we prove that if the first order deformations of the pair exist along certain first order deformations of the hypersurface $X_0$, then the twisted normal
bundle $N_{C_0/ X_0}(1) = N_{C_0 / X_0} \otimes \mathcal{O}_{\mathcal{P}^n} (1) \vert {}_{C_0}$ is generated by global sections.
</p>projecteuclid.org/euclid.ajm/1409168514_20140827154152Wed, 27 Aug 2014 15:41 EDTGeometry of isoparametric hypersurfaces in Riemannian manifoldshttp://projecteuclid.org/euclid.ajm/1409168515<strong>Jianquan Ge</strong>, <strong>Zizhou Tang</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 1, 117--126.</p><p><strong>Abstract:</strong><br/>
In our previous work, we studied isoparametric functions on Riemannian manifolds,
especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces
of a closed Riemannian manifold, there exists at least one minimal isoparametric hypersurface. In
this paper, we show such a minimal isoparametric hypersurface is also unique in the family if the
ambient manifold has positive Ricci curvature. Moreover, we give a proof of Theorem D claimed by
Q.M.Wang (without proof) which asserts that the focal submanifolds of an isoparametric function on
a complete Riemannian manifold are minimal. Further, we study isoparametric hypersurfaces with
constant principal curvatures in general Riemannian manifolds. It turns out that in this case the
focal submanifolds have the same properties as those in the standard sphere, i.e., the shape operator
with respect to any normal direction has common constant principal curvatures. Some necessary
conditions involving Ricci curvature and scalar curvature are also derived.
</p>projecteuclid.org/euclid.ajm/1409168515_20140827154152Wed, 27 Aug 2014 15:41 EDTA new curve algebraically but not rationally uniformized by radicalshttp://projecteuclid.org/euclid.ajm/1409168516<strong>Gian PietroI Pirola</strong>, <strong>Cecilia Rizzi</strong>, <strong>Enrico Schlesinger</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 1, 127--142.</p><p><strong>Abstract:</strong><br/>
We give a new example of a curve $C$ algebraically, but not rationally, uniformized by radicals. This means that $C$ has no map onto $\mathbb{P}^1$ with solvable Galois group, while there exists a curve
$C'$ that maps onto $C$ and has a finite morphism to $\mathbb{P}^1$ with solvable Galois group. We construct such a curve $C$ of genus $9$ in the second symmetric product of a general curve of genus
$2$. It is also an example of a genus $9$ curve that does not satisfy condition $S(4, 2, 9)$ of Abramovich and Harris.
</p>projecteuclid.org/euclid.ajm/1409168516_20140827154152Wed, 27 Aug 2014 15:41 EDTCrystalline and semi-stable representations in the imperfect residue field casehttp://projecteuclid.org/euclid.ajm/1409168517<strong>Kazuma Morita</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 1, 143--158.</p><p><strong>Abstract:</strong><br/>
Let $K$ be a $p$-adic local field with residue field $k$ such that $[k : k^p] = p^e \lt \infty$ and $V$ be a $p$-adic representation of $\mathrm{Gal}(\overline{K} / K)$. Then, by using the theory of $p$-adic differential
modules, we show that $V$ is a potentially crystalline (resp. potentially semi-stable) representation of $\mathrm{Gal}(\overline{K} / K)$ if and only if $V$ is a potentially crystalline (resp. potentially semi-stable)
representation of $\mathrm{Gal}(\overline{K^\mathrm{pf}} / K^\mathrm{pf})$ where $K^\mathrm{pf} / K$ is a certain $p$-adic local field whose residue field is the smallest perfect field $k^\mathrm{pf}$ containing
$k$. As an application, we prove the $p$-adic monodromy theorem of Fontaine in the imperfect residue field case.
</p>projecteuclid.org/euclid.ajm/1409168517_20140827154152Wed, 27 Aug 2014 15:41 EDTWarped product Einstein metrics over spaces with constant scalar curvaturehttp://projecteuclid.org/euclid.ajm/1409168518<strong>Chenxu He</strong>, <strong>Peter Petersen</strong>, <strong>William Wylie</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 1, 159--190.</p><p><strong>Abstract:</strong><br/>
In this paper we study warped product Einstein metrics over spaces with constant
scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is
isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension
of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of
solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product
Einstein metrics showing that the result is not true in dimension greater than three. We also give
some further natural curvature conditions that characterize the rigid examples in higher dimensions.
</p>projecteuclid.org/euclid.ajm/1409168518_20140827154152Wed, 27 Aug 2014 15:41 EDTDifferential Gerstenhaber algebras of generalized complex structureshttp://projecteuclid.org/euclid.ajm/1409168521<strong>Daniele Grandini</strong>, <strong>Yat-Sun Poon</strong>, <strong>Brian Rolle</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 2, 191--218.</p><p><strong>Abstract:</strong><br/>
Associated to every generalized complex structure is a differential Gerstenhaber
algebra (DGA). When the generalized complex structure deforms, so does the associated DGA. In
this paper, we identify the infinitesimal conditions when the DGA is invariant as the generalized
complex structure deforms. We prove that the infinitesimal condition is always integrable. When
the underlying manifold is a holomorphic Poisson nilmanifolds, or simply a group in the general,
and the geometry is invariant, we find a general construction to solve the infinitesimal conditions
under some geometric conditions. Examples and counterexamples of existence of solutions to the
infinitesimal conditions are given.
</p>projecteuclid.org/euclid.ajm/1409168521_20140827154202Wed, 27 Aug 2014 15:42 EDTCanonical maps of surfaces defined by abelian covershttp://projecteuclid.org/euclid.ajm/1409168522<strong>Rong Du</strong>, <strong>Yun Gao</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 2, 219--228.</p><p><strong>Abstract:</strong><br/>
In this paper, we classified the surfaces whose canonical maps are abelian covers over $\mathbb{P}^2. Moreover, we give defining equations for Perssson’s surface and Tan’s surfaces with odd canonical degrees explicitly.
</p>projecteuclid.org/euclid.ajm/1409168522_20140827154202Wed, 27 Aug 2014 15:42 EDTNon-nilpotent complex geometry of nilmanifolds and heterotic supersymmetryhttp://projecteuclid.org/euclid.ajm/1409168523<strong>Luis Ugarte</strong>, <strong>Raquel Villacampa</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 2, 229--246.</p><p><strong>Abstract:</strong><br/>
We classify non-nilpotent complex structures on 6-nilmanifolds and their associated
invariant balanced metrics. As an application we find a large family of solutions of the heterotic
supersymmetry equations with non-zero flux, non-flat instanton and constant dilaton satisfying the
anomaly cancellation condition with respect to the Chern connection.
</p>projecteuclid.org/euclid.ajm/1409168523_20140827154202Wed, 27 Aug 2014 15:42 EDTPostnikov-stability versus semistability of sheaveshttp://projecteuclid.org/euclid.ajm/1409168524<strong>Georg Hein</strong>, <strong>David Ploog</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 2, 247--262.</p><p><strong>Abstract:</strong><br/>
We present a novel notion of stable objects in a triangulated category. This Postnikov-stability is preserved by equivalences. We show that for the derived category of a projective variety this notion includes the case of
semistable sheaves. As one application we compactify a moduli space of stable bundles using genuine complexes via Fourier-Mukai transforms.
</p>projecteuclid.org/euclid.ajm/1409168524_20140827154202Wed, 27 Aug 2014 15:42 EDTComputing the walls associated to Bridgeland stability conditions on projective surfaceshttp://projecteuclid.org/euclid.ajm/1409168525<strong>Antony Maciocia</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 2, 263--280.</p><p><strong>Abstract:</strong><br/>
We derive constraints on the existence of walls for Bridgeland stability conditions for general projective surfaces. We show that in suitable planes of stability conditions the walls are bounded and derive conditions for when
the number of walls is globally finite. In examples, we show how to use the explicit conditions to locate walls and sometimes to show that there are no walls at all.
</p>projecteuclid.org/euclid.ajm/1409168525_20140827154202Wed, 27 Aug 2014 15:42 EDTSU(3)-holonomy metrics from nilpotent Lie groupshttp://projecteuclid.org/euclid.ajm/1409168526<strong>Diego Conti</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 2, 281--320.</p><p><strong>Abstract:</strong><br/>
One way of producing explicit Riemannian 6-manifolds with holonomy SU(3) is by integrating a flow of SU(2)-structures on a 5-manifold, called the hypo evolution flow. In this paper we classify invariant hypo SU(2)-structures
on nilpotent 5-dimensional Lie groups. We characterize the hypo evolution flow in terms of gauge transformations, and study the flow induced on the variety of frames on a Lie algebra taken up to automorphisms. We
classify the orbits of this flow for all hypo nilpotent structures, obtaining several families of cohomogeneity one metrics with holonomy contained in SU(3). We prove that these metrics cannot be extended to a complete
metric, unless they are flat.
</p>projecteuclid.org/euclid.ajm/1409168526_20140827154202Wed, 27 Aug 2014 15:42 EDTMini-walls for Bridgeland stability conditions on the derived category of sheaves over surfaceshttp://projecteuclid.org/euclid.ajm/1409168527<strong>Jason Lo</strong>, <strong>Zhenbo Qin</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 2, 321--344.</p><p><strong>Abstract:</strong><br/>
For the derived category of bounded complexes of sheaves on a smooth projective surface, Bridgeland and Arcara-Bertram constructed Bridgeland stability conditions $(Z_m , \mathcal{P}_m)$ parametrized by
$m \in (0, {+\infty})$. In this paper, we show that the set of mini-walls in $(0, {+\infty})$ of a fixed numerical type is locally finite. In addition, we strengthen a result of Bayer by proving that the moduli of polynomial
Bridgeland semistable objects of a fixed numerical type coincides with the moduli of $(Z_m , \mathcal{P}_m)$-semistable objects whenever $m$ is larger than a universal constant depending only on the numerical
type. We further identify the moduli of polynomial Bridgeland semistable objects with the Gieseker/Simpson moduli spaces and the Uhlenbeck compactification spaces.
</p>projecteuclid.org/euclid.ajm/1409168527_20140827154202Wed, 27 Aug 2014 15:42 EDTThe Euclid-Fourier-Mukai algorithm for elliptic surfaceshttp://projecteuclid.org/euclid.ajm/1409168528<strong>Marcello Bernardara</strong>, <strong>Georg Hein</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 2, 345--364.</p><p><strong>Abstract:</strong><br/>
We describe the birational correspondences, induced by the Fourier-Mukai functor, between moduli spaces of semistable sheaves on elliptic surfaces with sections, using the notion of $P$-stability in the derived category.
We give explicit conditions to determine whether these correspondences are isomorphisms. This is indeed not true in general and we describe the cases where the birational maps are Mukai flops. Moreover, this
construction provides examples of new compactifications of the moduli spaces of vector bundles via sheaves with torsion and via complexes. We finally get for any fixed dimension an isomorphism between the
Picard groups of the moduli spaces.
</p>projecteuclid.org/euclid.ajm/1409168528_20140827154202Wed, 27 Aug 2014 15:42 EDTEmbeddings of fields into simple algebras over global fieldshttp://projecteuclid.org/euclid.ajm/1409168529<strong>Sheng-Chi Shih</strong>, <strong>Tse-Chung Yang</strong>, <strong>Chia-Fu Yu</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 2, 365--386.</p><p><strong>Abstract:</strong><br/>
Let $F$ be a global field, $A$ a central simple algebra over $F$, and $K$ a finite (separable or not) field extension of $F$ with degree $[K : F]$ dividing the degree of $A$ over $F$. An embedding of $K$ into $A$ over
$F$ exists implies an embedding exists locally everywhere. In this paper we give detailed discussions about when the converse (i.e. the local-global principle in question) may hold.
</p>projecteuclid.org/euclid.ajm/1409168529_20140827154202Wed, 27 Aug 2014 15:42 EDTA mathematical theory of quantum sheaf cohomologyhttp://projecteuclid.org/euclid.ajm/1410186663<strong>Ron Donagi</strong>, <strong>Josh Guffin</strong>, <strong>Sheldon Katz</strong>, <strong>Eric Sharpe</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 3, 387--418.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to present a mathematical theory of the half-twisted $(0, 2)$ gauged linear sigma model and its correlation functions that agrees with and extends results from physics. The theory is associated
to a smooth projective toric variety $X$ and a deformation $\mathcal{E}$ of its tangent bundle $T_X$. It gives a quantum deformation of the cohomology ring of the exterior algebra of $\mathcal{E}*$. We prove that in
the general case, the correlation functions are independent of "nonlinear" deformations. We derive quantum sheaf cohomology relations that correctly specialize to the ordinary quantum cohomology relations described
by Batyrev in the special case $\mathcal{E} = T_X$.
</p>projecteuclid.org/euclid.ajm/1410186663_20140908103105Mon, 08 Sep 2014 10:31 EDTOn the injectivicy radius growth of complete noncompact Riemannian manifoldshttp://projecteuclid.org/euclid.ajm/1410186664<strong>Zhongyang Sun</strong>, <strong>Jianming Wan</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 3, 419--426.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce a global geometric invariant $\alpha(M)$ related to injectivity radius to complete non-compact Riemannian manifolds and prove: If $\alpha(M^n) \gt 1$, then $M^n$ is isometric
to $\mathbb{R}^n$ when Ricci curvature is non-negative, and is diffeomorphic to $\mathbb{R}^n$ for $n \neq 4$ and homeomorphic to $\mathbb{R}^4$ for $n = 4$ if without any curved assumption.
</p>projecteuclid.org/euclid.ajm/1410186664_20140908103105Mon, 08 Sep 2014 10:31 EDTImprovements of the Five Halves Theorem of J. Boardman with respect to the decomposability degreehttp://projecteuclid.org/euclid.ajm/1410186665<strong>Patricia E. Desideri</strong>, <strong>Pedro L. Q. Pergher</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 3, 427--438.</p><p><strong>Abstract:</strong><br/>
Let $(M^m, T)$ be a smooth involution on a closed smooth $m$-dimensional manifold and $F = {\bigcup}^n_{j=0} F^j (n \lt m)$ its fixed point set, where $F^j$ denotes the union of those components of
$F$ having dimension $j$. The famous Five Halves Theorem of J. Boardman, announced in 1967, establishes that, if $F$ is nonbounding, then $m \leq \frac{5}{2} n$; further, this estimative is best possible.
In this paper we obtain improvements of this theorem, taking into account certain natural numbers which we call the decomposability degrees $\ell(F^j)$ of the nonbounding components $F^j$ of $F$ (see the
definition in Section 1). Also, these improvements are obtained under assumptions on the set of dimensions occurring in $F$, which we denote $\pi_0(F)$. The main result of this paper is: suppose the
involution $(M^m, T)$ has $\pi_0(F) = \{ 0, 1, \dots, j, n \}$, where $2 \leq j \lt n \lt m$ and $F^j$ is nonbounding. Write $\mathcal{M}(n - j)$ for the function of $n - j$ defined in the following way:
writing $n - j = 2^p q$, where $q \geq 1$ is odd and $p \geq 0$, $M(n - j) = 2n + p - q + 1$ if $p \leq q$ and $M(n - j) = 2n + 2^{p-q}$ if $p \geq q$. Then $m \leq \mathcal{M}(n - j) + 2j + \ell (F^j)$. In addition, we
develop a method to construct involutions $(M^m, T)$ with $\pi_0(F)$ as above, in some special situations, which in some cases will show that the above bound is best possible. This will provide some improvements
of the general Five Halves Theorem $(\pi_0(F) = \{ i / 0 \leq i \leq n \} )$, by considering the particular case $j = n - 1$.
</p>projecteuclid.org/euclid.ajm/1410186665_20140908103105Mon, 08 Sep 2014 10:31 EDTOn certain generalized Hardy's inequalities and applicationshttp://projecteuclid.org/euclid.ajm/1410186666<strong>Demetrios A. Pliakis</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 3, 439--464.</p>projecteuclid.org/euclid.ajm/1410186666_20140908103105Mon, 08 Sep 2014 10:31 EDTStable logarithmic maps to Deligne-Faltings pairs IIhttp://projecteuclid.org/euclid.ajm/1410186667<strong>Dan Abramovich</strong>, <strong>Qile Chen</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 3, 465--488.</p><p><strong>Abstract:</strong><br/>
We make an observation which enables one to deduce the existence of an algebraic stack of logarithmic maps for all generalized Deligne-Faltings logarithmic structures (in particular simple normal crossings divisors) from
the simplest case with characteristic generated by $\mathbb{N}$ (essentially the smooth divisor case).
</p>projecteuclid.org/euclid.ajm/1410186667_20140908103105Mon, 08 Sep 2014 10:31 EDTA holographic principle for the existence of parallel Spinor fields and an inequality of Shi-Tam typehttp://projecteuclid.org/euclid.ajm/1410186668<strong>Oussama Hijazi</strong>, <strong>Sebastián Montiel</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 3, 489--506.</p><p><strong>Abstract:</strong><br/>
Suppose that $\Sigma = \partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $(M, \langle , \rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature
$H$ of $\Sigma$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric $\langle , \rangle {}_H = H^2 \langle , \rangle$ is at least $n/2$ and equality holds
if and only if there exists a non-trivial parallel spinor field on $M$. As a consequence, if $\Sigma$ admits an isometric and isospin immersion $F$ with mean curvature $H_0$ as a hypersurface into another spin
Riemannian manifold $M_0$ admitting a parallel spinor field, then$$\int_{\Sigma} H{ } d\Sigma \leq \int_{\Sigma} \frac{H^2_0}{H} { } d\Sigma$$where $H$ is the mean curvature of $\Sigma$ as the boundary
of $M$ and $H_0$ stands for the mean curvature of the immersion $F$ of $\Sigma$ into $\mathbb{R}^{n+1}$. Equality holds if and only if $\Sigma$ is connected, $M$ is a Euclidean domain and the embedding
of $\Sigma$ in $M$ and its immersion in $\mathbb{R}^{n+1}$ are congruent.
</p>projecteuclid.org/euclid.ajm/1410186668_20140908103105Mon, 08 Sep 2014 10:31 EDTAsymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaceshttp://projecteuclid.org/euclid.ajm/1410186669<strong>Robin De Jong</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 3, 507--524.</p><p><strong>Abstract:</strong><br/>
Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical
invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as
a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of
holomorphic differentials. In this paper we determine precise formulas, up to and including constant
terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann
surfaces. As a corollary we state precise asymptotic formulas for the beta-invariant introduced
around 2000 by R. Hain and D. Reed. These formulas are a refinement of a result Hain and Reed
prove in their paper. We illustrate our results with some explicit calculations on degenerating genus
two surfaces.
</p>projecteuclid.org/euclid.ajm/1410186669_20140908103105Mon, 08 Sep 2014 10:31 EDTSymmetry defect of algebraic varietieshttp://projecteuclid.org/euclid.ajm/1410186670<strong>S. Janeczko</strong>, <strong>Z. Jelonek</strong>, <strong>M. A. S. Ruas</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 3, 525--544.</p><p><strong>Abstract:</strong><br/>
Let $X, Y \subset k^m(k = \mathbb{R},\mathbb{C})$ be smooth manifolds. We investigate the central symmetry of the configuration of $X$ and $Y$. For $p \in k^m$ we introduce a number $\mu(p)$ of pairs of
points $x \in X$ and $y \in Y$ such that $p$ is the center of the interval $\overline{xy}$. We show that if $X, Y$ (including the case $X = Y$ ) are algebraic manifolds in a general position, then there is a closed
(semi-algebraic) set $B \subset k^m$, called symmetry defect set of the $X$ and $Y$ configuration, such that the function $\mu$ is locally constant and not identically zero outside $B$. If $k = \mathbb{C}$, we
estimate the number $\mu$ (in fact we compute it in many cases) and show that the symmetry defect is an algebraic hypersurface and consequently the function $\mu$ is constant and positive outside $B$. We also
show that in the generic case the topological type of the symmetry defect set of a plane curve is constant, i.e. the symmetry defect sets for two generic curves of the same degree are homeomorphic (by the same
method we can prove similar statement for any irreducible family of smooth varieties $Z^n \subset \mathbb{C}^{2n}$). Moreover, for $k = \mathbb{R}$, we estimate the number of connected components of the
set $U = k^m \backslash B$. In the last section we give an algorithm to compute the symmetry defect set for complex smooth affine varieties in general position.
</p>projecteuclid.org/euclid.ajm/1410186670_20140908103105Mon, 08 Sep 2014 10:31 EDTStructure of Hochschild cohomology of path algebras and differential formulation of Euler's polyhedron formulahttp://projecteuclid.org/euclid.ajm/1410186671<strong>Li Guo</strong>, <strong>Fang Li</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 3, 545--572.</p><p><strong>Abstract:</strong><br/>
This article studies the Lie algebra $\mathrm{Der}(\mathrm{k}\Gamma)$ of derivations on the path algebra $\mathrm{k}\Gamma$ of a quiver $\Gamma$ and the Lie algebra on the first Hochschild cohomology
group $HH1(\mathrm{k}\Gamma)$. We relate these Lie algebras to the algebraic and combinatorial properties of the path algebra. Characterizations of derivations on a path algebra are obtained, leading to a canonical
basis of $\mathrm{Der}(\mathrm{k}\Gamma)$ and its Lie algebra properties. Special derivations are associated to the vertices, arrows and faces of a quiver, and the concepts of a connection matrix and boundary
matrix are introduced to study the relations among these derivations, giving rise to an interpretation of Euler's polyhedron formula in terms of derivations. By taking dimensions, this relation among spaces of derivations
recovers Euler's polyhedron formula. This relation also leads to a combinatorial construction of a canonical basis of the Lie algebra $HH1(\mathrm{k}\Gamma)$, together with a new semidirect sum decomposition
of $HH1(\mathrm{k}\Gamma)$.
</p>projecteuclid.org/euclid.ajm/1410186671_20140908103105Mon, 08 Sep 2014 10:31 EDTProjective completions of affine varieties via degree-like functionshttp://projecteuclid.org/euclid.ajm/1415284978<strong>Pinaki Mondal</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 4, 573--602.</p><p><strong>Abstract:</strong><br/>
We study projective completions of affine algebraic varieties induced by filtrations on their coordinate rings. In particular, we study the effect of the “multiplicative” property of filtrations on the corresponding completions
and introduce a class of projective completions (of arbitrary affine varieties) which generalizes the construction of toric varieties from convex rational polytopes. As an application we recover (and generalize to varieties
over algebraically closed fields of arbitrary characteristics) a “finiteness” property of divisorial valuations over complex affine varieties proved in “Divisorial valuations via arcs” [T. de Fernex, L. Ein, and S. Ishii, Publ.
Res. Inst. Math. Sci. , 44:2 (2008), pp. 425–448]. We also find a formula for the pull-back of the “divisor at infinity” and apply it to compute the matrix of intersection numbers of the curves at infinity on a class
of compactifications of certain affine surfaces.
</p>projecteuclid.org/euclid.ajm/1415284978_20141106094300Thu, 06 Nov 2014 09:43 ESTA result on Ricci curvature and the second Betti numberhttp://projecteuclid.org/euclid.ajm/1415284979<strong>Jianming Wan</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 4, 603--608.</p><p><strong>Abstract:</strong><br/>
We prove that the second Betti number of a compact Riemannian manifold vanishes under certain Ricci curved restriction. As consequences we obtain an interesting curved restriction for compact
Kähler-Einstein manifolds and a homology sphere theorem in ${\rm dim}=4,5$.
</p>projecteuclid.org/euclid.ajm/1415284979_20141106094300Thu, 06 Nov 2014 09:43 ESTSmall four-manifolds without non-singular solutions of normalized Ricci flowshttp://projecteuclid.org/euclid.ajm/1415284980<strong>Masashi Ishida</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 4, 609--622.</p><p><strong>Abstract:</strong><br/>
It is known that connected sums $X\# K 3 \# (\Sigma_g \times \Sigma_h) \# \ell_1 (S^1 \times S^3) \# \ell_2 \overline{\mathbb{C}P^2}$ satisfy the Gromov-Hitchin-Thorpe type inequality, but can not admit non-singular
solutions of the normalized Ricci flow for any initial metric, where $\Sigma_g \times \Sigma_h$ is the product of two Riemann surfaces of odd genus, $\ell_1, \ell_2 \gt 0$ are sufficiently large positive
integers, $g, h \gt 3$ are also sufficiently large positive odd integers, and $X$ is a certain irreducible symplectic 4-manifold. These examples are closely related with a conjecture of Fang, Zhang and Zhang.
In the current article, we point out that there still exist 4-manifolds with the same property even if $\ell_1 = \ell_2 = 0$ and $g = h = 3$. The topology of these new examples are smaller than that of previously known examples.
</p>projecteuclid.org/euclid.ajm/1415284980_20141106094300Thu, 06 Nov 2014 09:43 ESTHypoellipticity of the $\overline{\partial}$-Neumann problem at a point of infinite typehttp://projecteuclid.org/euclid.ajm/1415284981<strong>Luca Baracco</strong>, <strong>Tran Vu Khanh</strong>, <strong>Giuseppe Zampieri</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 4, 623--632.</p><p><strong>Abstract:</strong><br/>
We prove local hypoellipticity of the complex Laplacian $\square$ in a domain which has superlogarithmic estimates outside a curve transversal to the CR directions and for which the holomorphic tangential
derivatives of a defining function are superlogarithmic multipliers in the sense of "A general method of weights in the $\overline{\partial}$-Neumann problem," [T. V. Khanh, Ph.D. Thesis, Padua (2009)].
</p>projecteuclid.org/euclid.ajm/1415284981_20141106094300Thu, 06 Nov 2014 09:43 ESTAsymptotic spectral flow for Dirac operators of disjoint Dehn twistshttp://projecteuclid.org/euclid.ajm/1415284982<strong>Chung-Jun Tsai</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 4, 633--686.</p><p><strong>Abstract:</strong><br/>
Let $Y$ be a compact, oriented 3-manifold with a contact form $a$. For any Dirac operator $\mathcal{D}$, we study the asymptotic behavior of the spectral flow between $\mathcal{D}$ and
$\mathcal{D} + \mathrm{cl}(-\frac{ir}{2}a)$ as $r \to \infty$. If $a$ is the Thurston-Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the next order
term of the spectral flow function is $\mathcal{O}(r)$.
</p>projecteuclid.org/euclid.ajm/1415284982_20141106094300Thu, 06 Nov 2014 09:43 ESTBoundaries of cycle spaces and degenerating Hodge structureshttp://projecteuclid.org/euclid.ajm/1415284983<strong>Tatsuki Hayama</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 4, 687--706.</p><p><strong>Abstract:</strong><br/>
We study a property of cycle spaces in connection with degenerating Hodge structures of odd-weight, and we construct maps from some partial compactifications of period domains to the Satake compatifications
of Siegel spaces. These maps are a generalization of the maps from the toroidal compactifications of Siegel spaces to the Satake compactifications. We also show continuity of these maps for the case for the Hodge
structure of Calabi-Yau threefolds with $h^{2,1} = 1$.
</p>projecteuclid.org/euclid.ajm/1415284983_20141106094300Thu, 06 Nov 2014 09:43 ESTCM elliptic curves and primes captured by quadratic polynomialshttp://projecteuclid.org/euclid.ajm/1415284984<strong>Qingzhong Ji</strong>, <strong>Hourong Qin</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 4, 707--726.</p><p><strong>Abstract:</strong><br/>
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with complex multiplication. For a prime $p$, some formulas for $a_p = p + 1 \sharp E(\mathbb{F}_p)$ are given in terms of the binomial coefficients.
We show that the equality $a_p = r$ holds for some fixed integer $r$ if and only if a certain quadratic polynomial represents the prime $p$. In particular, for $E \colon y^2 = x^3 + x, a_p = 2$ holding for an odd prime
$p$ if and only if $p$ is of the form $n^2 + 1$ and for $E \colon y^2 = x^3 - 11x + 14, a_p = 2$ holding for an odd prime $p$ if and only if $p$ is of the form $(4n)^2 + 1; a_p = -2$ holding for an odd prime $p$ if and
only if $p$ is of the form $(4n + 2)^2 + 1$. In some CM cases the Lang-Trotter conjecture and the Hardy-Littlewood conjecture are equivalent.
</p>projecteuclid.org/euclid.ajm/1415284984_20141106094300Thu, 06 Nov 2014 09:43 ESTA no breathers theorem for some noncompact Ricci flowshttp://projecteuclid.org/euclid.ajm/1415284985<strong>Qi S. Zhang</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 4, 727--756.</p><p><strong>Abstract:</strong><br/>
Under suitable conditions near infinity and assuming boundedness of curvature
tensor, we prove a no breathers theorem in the spirit of Ivey-Perelman for some noncompact Ricci
flows. These include Ricci flows on asymptotically flat (AF) manifolds with positive scalar curvature,
which was studied in "Mass under the Ricci flow," [X. Dai and L. Ma, Comm. Math. Phys. 274:1 (2007), pp. 65–80] and "Rotationally symmetric Ricci flow on asymptotically
flat manifolds," [T. A. Oliynyk, and E. Woolgar, Comm. Anal. Geom. , 15:3 (2007), pp. 535–568] in connection with general relativity. Since the method for the
compact case faces a difficulty, the proof involves solving a new non-local elliptic equation which is
the Euler-Lagrange equation of a scaling invariant log Sobolev inequality.
It is also shown that the Ricci flow on AF manifolds with positive scalar curvature is uniformly $\kappa$
noncollapsed for all time. This result, being different from Perelman’s local noncollapsing result
which holds in finite time, seems to have implications for the issue of longtime convergence.
</p>projecteuclid.org/euclid.ajm/1415284985_20141106094300Thu, 06 Nov 2014 09:43 EST$\mathcal{F}$-stability for self-shrinking solutions to mean curvature flowhttp://projecteuclid.org/euclid.ajm/1417489241<strong>Ben Andrews</strong>, <strong>Haizhong Li</strong>, <strong>Yong Wei</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 5, 757--778.</p><p><strong>Abstract:</strong><br/>
In this paper, we formulate the notion of the $\mathcal{F}$-stability of self-shrinking solutions to mean curvature flow in arbitrary codimension. Then we give some classifications of the $\mathcal{F}$-stable self-shrinkers
in arbitrary codimension. We show that the only $\mathcal{F}$-stable self-shrinking solution which is a closed minimal submanifold in a sphere must be the shrinking sphere. We also prove that the spheres and planes
are the only $\mathcal{F}$-stable self-shrinkers with parallel principal normal. In the codimension one case, our results reduce to those of Colding and Minicozzi.
</p>projecteuclid.org/euclid.ajm/1417489241_20141201220046Mon, 01 Dec 2014 22:00 ESTDirac Lie Groupshttp://projecteuclid.org/euclid.ajm/1417489242<strong>David Li-Bland</strong>, <strong>Eckhard Meinrenken</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 5, 779--816.</p><p><strong>Abstract:</strong><br/>
A classical theorem of Drinfel'd states that the category of simply connected Poisson Lie groups $H$ is isomorphic to the category of Manin triples $(\mathfrak{d, g, h})$, where $\mathfrak{h}$ is the Lie algebra of $H$.
In this paper, we consider Dirac Lie groups, that is, Lie groups $H$ endowed with a multiplicative Courant algebroid $A$ and a Dirac structure $E \subseteq \mathbb{A}$ for which the multiplication is a Dirac
morphism. It turns out that the simply connected Dirac Lie groups are classified by so-called Dirac Manin triples. We give an explicit construction of the Dirac Lie group structure defined by a Dirac Manin triple,
and develop its basic properties.
</p>projecteuclid.org/euclid.ajm/1417489242_20141201220046Mon, 01 Dec 2014 22:00 ESTIrreducible quasifinite modules over a class of Lie algebras of block typehttp://projecteuclid.org/euclid.ajm/1417489243<strong>Hongjia Chen</strong>, <strong>Xiangqian Guo</strong>, <strong>Kaiming Zhao</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 5, 817--828.</p><p><strong>Abstract:</strong><br/>
For any nonzero complex number $q$, there is a Lie algebra of Block type, denoted by $\mathcal{B}(q)$. In this paper, a complete classification of irreducible quasifinite modules is given. More precisely, an irreducible
quasifinite module is a highest weight or lowest weight module, or a module of intermediate series. As a consequence, a classification for uniformly bounded modules over another class of Lie algebras, the semi-direct
product of the Virasoro algebra and a module of intermediate series, is also obtained. Our method is conceptional, instead of computational.
</p>projecteuclid.org/euclid.ajm/1417489243_20141201220046Mon, 01 Dec 2014 22:00 ESTPeriodic constant mean curvature surfaces in $\mathbb{H}^2 \times \mathbb{R}$http://projecteuclid.org/euclid.ajm/1417489244<strong>Laurent Mazet</strong>, <strong>M. Magdalena Rodríguez</strong>, <strong>Harold Rosenberg</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 5, 829--858.</p>projecteuclid.org/euclid.ajm/1417489244_20141201220046Mon, 01 Dec 2014 22:00 ESTExistence of approximate Hermitian-Einstein structures on semi-stable bundleshttp://projecteuclid.org/euclid.ajm/1417489245<strong>Adam Jacob</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 5, 859--884.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle $E$ over a compact Kähler manifold $X$. It is shown that if $E$ is semi-stable, then Donaldson’s functional is bounded from below.
This implies that $E$ admits an approximate Hermitian-Einstein structure, generalizing a classic result of Kobayashi for projective manifolds to the Kähler case. As an application some basic properties of semi-stable
vector bundles over compact Kähler manifolds are established, such as the fact that semi-stability is preserved under certain exterior and symmetric products.
</p>projecteuclid.org/euclid.ajm/1417489245_20141201220046Mon, 01 Dec 2014 22:00 ESTTame Fréchet structures for affine Kac-Moody groupshttp://projecteuclid.org/euclid.ajm/1417489246<strong>Walter Freyn</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 5, 885--928.</p><p><strong>Abstract:</strong><br/>
We construct holomorphic loop groups and their associated affine Kac-Moody groups and prove that they are tame Fréchet manifolds; furthermore we study the adjoint action of these groups. These results form the functional
analytic core for a theory of affine Kac-Moody symmetric spaces, that will be developed in forthcoming papers. Our construction also solves the problem of complexification of completed Kac-Moody groups: we obtain a
description of complex completed Kac-Moody groups and, using this description, deduce constructions of their non-compact real forms.
</p>projecteuclid.org/euclid.ajm/1417489246_20141201220046Mon, 01 Dec 2014 22:00 ESTOn the estimate of the first positive eigenvalue of a sublaplacian in a pseudohermitian manifoldhttp://projecteuclid.org/euclid.ajm/1417489247<strong>Yen-Wen Fan</strong>, <strong>Ting-Jung Kuo</strong>. <p><strong>Source: </strong>Asian Journal of Mathematics, Volume 18, Number 5, 929--946.</p><p><strong>Abstract:</strong><br/>
In this paper, we first obtain a CR version of Yau’s gradient estimate for eigenfunctions of a sublaplacian. Second, by using CR analogue of Li-Yau’s eigenvalue estimate, we are able to obtain a lower bound of the first
positive eigenvalue in a pseudohermitian manifold of nonvanishing pseudohermitian torsion and nonpositive lower bound on pseudohermitian Ricci curvature.
</p>projecteuclid.org/euclid.ajm/1417489247_20141201220046Mon, 01 Dec 2014 22:00 EST