Showing papers in "Pacific Journal of Mathematics in 2017"
TL;DR: In this article, a family of geometric flows introduced by Bourguignon in [3] that generalize the Ricci flow is presented. And the authors prove short time existence and provide curvature estimates.
Abstract: In this paper we present some results on a family of geometric flows introduced by Bourguignon in [3] that generalize the Ricci flow. For suitable values of the scalar parameter involved in these flows, we prove short time existence and provide curvature estimates. We also state some results on the associated solitons. CONTENTS
68 citations
TL;DR: It is shown that as r increases, the Vietoris-Rips complex of thecircle obtains the homotopy types of the circle, the 3-sphere, the 5-spheric, the 7-spheres, ..., until finally it is contractible.
Abstract: Given a metric space X and a distance threshold r>0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambient Cech complex of the circle also obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible.
54 citations
TL;DR: In this article, the complex Hessian equation on closed Hermitian manifolds was solved using technical tools in \cite{TW5] and used to generalize the Kahler case.
Abstract: In this paper, using the technical tools in \cite{TW5}, we solve the complex Hessian equation on closed Hermitian manifolds, which generalizes the the Kahler case results in \cite{HMW} and \cite{DK}.
40 citations
TL;DR: In this article, the notion of Einstein-type structure on a Riemannian manifold was introduced, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe Solitons and quasi-Einstein manifolds.
Abstract: In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold $\varrg$, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen.
37 citations
TL;DR: In this article, it was shown that the gap from regularity to smoothness is logarithmic in nature, but the smallness depends on a dimensionless quantity of the initial data.
Abstract: Smooth solutions to the axi-symmetric Navier-Stokes equations obey the following maximum principle: $$\sup_{t\geq 0}\|rv^\theta(t, \cdot)\|_{L^\infty} \leq \|rv^\theta(0, \cdot)\|_{L^\infty}.$$ We prove that all solutions with initial data in $H^{\frac{1}{2}}$ is smooth globally in time if $rv^\theta$ satisfies a kind of Form Boundedness Condition (FBC) which is invariant under the natural scaling of the Navier-Stokes equations. In particular, if $rv^\theta$ satisfies \begin{equation}
onumber \sup_{t \geq 0}|rv^\theta(t, r, z)| \leq C_\ast|\ln r|^{- 2},\ \ r \leq \delta_0 \in (0, \frac{1}{2}),\ C_\ast < \infty, \end{equation} then our FBC is satisfied. Here $\delta_0$ and $C_\ast$ are independent of neither the profile nor the norm of the initial data. So the gap from regularity is logarithmic in nature. We also prove the global regularity of solutions if $\|rv^\theta(0, \cdot)\|_{L^\infty}$ or $\sup_{t \geq 0}\|rv^\theta(t, \cdot)\|_{L^\infty(r \leq r_0)}$ is small but the smallness depends on certain dimensionless quantity of the initial data.
36 citations
TL;DR: In this paper, a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds) is presented, which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's splitting theorem.
Abstract: We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's splitting theorem. Our approach turns out to be essentially canonical, and as a byproduct, we obtain an equivariant version of the latter theorem.
34 citations
TL;DR: In this article, the authors consider the problem of determining when a theta representation has exactly a one-dimensional space of Whittaker functionals, in which case it is distinguished.
Abstract: For Brylinski-Deligne covering groups of an arbitrary split reductive group, we consider theta representations attached to certain exceptional genuine characters. The goal of the paper is to determine when a theta representation has exactly a one-dimensional space of Whittaker functionals, in which case it is called distinguished. For this purpose, we first give effective lower and upper bounds for the dimension of Whittaker functionals for general theta representations. As a consequence, the dimension in many cases can be reduced to simple combinatorial computations, e.g., the Kazhdan-Patterson covering groups, or covering groups whose complex dual group is of adjoint type. In the second part of the paper, we consider covering groups of certain simply-connected groups and give necessary and sufficient condition for the theta representation to be distinguished. There are subtleties arising from the relation between the rank and the degree of the covering group. However, in each case we will determine the exceptional character such that its associated theta representation is distinguished.
33 citations
TL;DR: In this article, a Van Est map for representations up to homotopy on 2-term graded vector bundles has been obtained for Lie groupoids with k-homogeneous cochains.
Abstract: VB-groupoids define a special class of Lie groupoids which carry a compatible linear structure. In this paper, we show that their differentiable cohomology admits a refinement by considering the complex of cochains which are k-homogeneous on the linear fiber. Our main result is a Van Est theorem for such cochains. We also work out two applications to the general theory of representations of Lie groupoids and algebroids. The case k=1 yields a Van Est map for representations up to homotopy on 2-term graded vector bundles. Arbitrary k-homogeneous cochains on suitable VB-groupoids lead to a novel Van Est theorem for differential forms on Lie groupoids with values in a representation.
29 citations
TL;DR: In this paper, it was shown that the heart of a cosilting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption.
Abstract: In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption. Moreover, in the cosilting case the previous conditions are related to the coaisle of the t-structure being a definable subcategory. If we further assume our triangulated category to be algebraic, it follows that the heart of any nondegenerate compactly generated t-structure is a Grothendieck category.
27 citations
TL;DR: In this article, the growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions are established for Clifford algebras, based on a new convex combination identity.
Abstract: The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc $\\mathbb D\\subset \\mathbb C$ in the setting of Clifford algebras, based on a new convex combination identity. The analogous results are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems fail in general and hold only for some subclasses with the starlike or convex assumption.
26 citations
TL;DR: In this article, the Molino theory for equicontinuous matchbox manifolds is extended to the case of totally disconnected transversals, which is a special case of weak solenoids.
Abstract: A matchbox manifold is a foliated space with totally disconnected transversals, and an equicontinuous matchbox manifold is the generalization of Riemannian foliations for smooth manifolds in this context. In this paper, we develop the Molino theory for all equicontinuous matchbox manifolds. Our work extends the Molino theory developed in the work of Alvarez Lopez and Moreira Galicia which required the hypothesis that the holonomy actions for these spaces satisfy the strong quasi-analyticity condition. The methods of this paper are based on the authors' previous works on the structure of weak solenoids, and provide many new properties of the Molino theory for the case of totally disconnected transversals, and examples to illustrate these properties. In particular, we show that the Molino space need not be uniquely well-defined, unless the global holonomy dynamical system is tame, a notion defined in this work. We show that examples in the literature for the theory of weak solenoids provide examples for which the strong quasi-analytic condition fails. Of particular interest is a new class of examples of equicontinuous minimal Cantor actions by finitely generated groups, whose construction relies on a result of Lubotzky. These examples have non-trivial Molino sequences, and other interesting properties.
TL;DR: In this paper, the authors describe the presentation of braid groups of type ADE and show how these presentations are compatible with mutation of quivers, building on work of Barot and Marsh for Coxeter groups, and then give a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras with potential.
Abstract: We describe presentations of braid groups of type ADE and show how these presentations are compatible with mutation of quivers, building on work of Barot and Marsh for Coxeter groups. In types A and D these presentations can be understood geometrically using triangulated surfaces. We then give a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras of quivers with potential.
TL;DR: For a Legendrian torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and Kalman as discussed by the authors showed that these exact Lagrangian fillings are pairwise non-isotopic.
Abstract: For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and Kalman constructed $C_n$ exact Lagrangian fillings, where $C_n$ is the $n$-th Catalan number. We show that these exact Lagrangian fillings are pairwise non-isotopic through exact Lagrangian isotopy. To do that, we compute the augmentations induced by the exact Lagrangian fillings $L$ to $\mathbb{Z}_2[H_1(L)]$ and distinguish the resulting augmentations.
TL;DR: In this paper, the ideals of linear operators between Banach spaces determined by the transformation of vector-valued sequences involving the new sequence space introduced by Karn and Sinha \cite{karnsinha} and the classical spaces of absolutely, weakly and Cohen strongly summable sequences are studied.
Abstract: We study the ideals of linear operators between Banach spaces determined by the transformation of vector-valued sequences involving the new sequence space introduced by Karn and Sinha \cite{karnsinha} and the classical spaces of absolutely, weakly and Cohen strongly summable sequences. As applications, we prove a new factorization theorem for absolutely summing operators and a contribution to the existence of infinite dimensional spaces formed by non-absolutely summing operators is given.
TL;DR: The tensor product decomposition of the half of universal enveloping algebra associated with a Weyl group element was proved in this paper, which was conjectured by Berenstein and Greenstein.
Abstract: We prove the tensor product decomposition of the half of quantized universal enveloping algebra associated with a Weyl group element which was conjectured by Berenstein and Greenstein.
TL;DR: In this article, it was shown that any compactly generated t-structure in the derived category of a commutative noetherian ring has a heart which is equivalent to the category of finitely presented objects of some locally coherent Grothendieck category.
Abstract: We show that, under particular conditions, if a t-structure in the unbounded derived category of a locally coherent Grothendieck category restricts to the bounded derived category of its category of finitely presented objects, then its heart is itself a locally coherent Grothendieck category. Those particular conditions are always satisfied when the Grothendieck category is arbitrary and one considers the t-structure associated to a torsion pair in the category of finitely presented objects. They are also satisfied when one takes any compactly generated t-structure in the derived category of a commutative noetherian ring which restricts to the bounded derived category of finitely generated modules. As a consequence, any t-structure in this latter bounded derived category has a heart which is equivalent to the category of finitely presented objects of some locally coherent Grothendieck category.
TL;DR: In this article, the irreducible components of the moduli space of instanton sheaves on the rank 2 torsion free sheaves were studied, where the modulus space of stable sheaves was studied.
Abstract: We study the irreducible components of the moduli space of instanton sheaves on $\mathbb{P}^3$, that is rank 2 torsion free sheaves $E$ with $c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we classify all instanton sheaves with $c_2(E)\le4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space ${\mathcal T}(d)$ of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $P(t)=d\cdot t$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we describe all the irreducible components of ${\mathcal T}(d)$ for $d\le4$.
TL;DR: In this article, the authors investigated closure results for certain classes of groups with invariant length functions, including weakly sofic groups and hyperlinear groups, and showed that the restricted standard wreath product of two such groups is Ω(C$-approximable).
Abstract: We investigate closure results for $\C$-approximable groups, for certain classes $\C$ of groups with invariant length functions. In particular we prove, each time for certain (but not necessarily the same) classes $\C$ that: \linebreak (i) the direct product of two $\C$-approximable groups is $\C$-approximable; (ii) the restricted standard wreath product $G \wr H$ is $\C$-approximable when $G$ is $\C$-approximable and $H$ is residually finite; and (iii) a group $G$ with normal subgroup $N$ is $\C$-approximable when $N$ is $\C$-approximable and $G/N$ is amenable. Our direct product result is valid for LEF, weakly sofic and hyperlinear groups, as well as for all groups that are approximable by finite groups equipped with commutator-contractive invariant length functions (considered in \cite{Thom}). Our wreath product result is valid for weakly sofic groups, and we prove it separately for sofic groups. Our result on extensions by amenable groups is valid for weakly sofic groups, and was proved in \cite[Theorem 1 (3)]{ElekSzabo} for sofic groups $N$. \
TL;DR: In this article, a weighted Sobolev estimate of the Bergman projection on the Hartogs triangle is given, where the weight is some power of the distance to the singularity at the boundary.
Abstract: We prove a weighted Sobolev estimate of the Bergman projection on the Hartogs triangle, where the weight is some power of the distance to the singularity at the boundary. This method also applies to the $n$-dimensional generalization of the Hartogs triangle.
TL;DR: All articles published by MSP become open access after five years past publication (meaning on the fifth January 1st after the publication date).
Abstract: Click on the DOI link to access the article (may not be free).
All articles published by MSP become open access after five years past publication (meaning on the fifth January 1st after the publication date).
TL;DR: In this article, Li-Yau-Hamilton-type Harnack estimates for positive solutions of Fisher's equation were derived and used to obtain lower bounds on the speed of traveling wave solutions.
Abstract: In this paper, we derive several differential Harnack estimates (also known as Li-Yau-Hamilton-type estimates) for positive solutions of Fisher's equation. We use the estimates to obtain lower bounds on the speed of traveling wave solutions and to construct classical Harnack inequalities.
TL;DR: It turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality, which answers a question posed by N. Katz.
Abstract: An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a product formula, in terms of congruence considerations involving that polynomial, for the size of such an isogeny class. In this paper, we give a new, transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality; this answers a question posed by N. Katz in [11, Remark 8.7].
TL;DR: In this article, the authors introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order.
Abstract: We introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order. We also study the convergence of the discrete curvature flow. Discrete curvature flow of second order is an analogue of smooth Ricci flow.