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 EDTOn 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 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 EDT