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Tian-Jun Li

Publications and source records attributed to Tian-Jun Li.

At least 55 records · Page 3Linked to original sources
arXiv2012

Incompressible Navier-Stokes Equations from Einstein Gravity with Chern-Simons Term

In (2+1)-dimensional hydrodynamic systems with broken parity, the shear and bulk viscosity is joined by the Hall viscosity and curl viscosity. The dual holographic model has been constructed by coupling a pseudo scalar to the gravitational Chern-Simons term in (3+1)-dimensional bulk gravity. In this paper, we investigate the non-relativistic fluid with Hall viscosity and curl viscosity living on a finite radial cutoff surface in the bulk. Employing the non-relativistic hydrodynamic expansion method, we obtain the incompressible Navier-Stokes equations with Hall viscosity and curl viscosity. Unlike the shear viscosity, the ratio of the Hall viscosity over entropy density is found to be cutoff scale dependent, and it tends to zero when the cutoff surface approaches to the horizon of the background spacetime.

hep-th↗
arXiv2012

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$. In a different direction, we prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered then the 4-manifold is virtually symplectic unless its virtual first Betti number is 1.

math.GT↗
arXiv2012

Almost Kähler structures on four dimensional unimodular Lie algebras

Let $J$ be an almost complex structure on a 4-dimensional and unimodular Lie algebra $\mathfrak{g}$. We show that there exists a symplectic form taming $J$ if and only if there is a symplectic form compatible with $J$. We also introduce groups $H^+_J(\mathfrak{g})$ and $H^-_J(\mathfrak{g})$ as the subgroups of the Chevalley-Eilenberg cohomology classes which can be represented by $J$-invariant, respectively $J$-anti-invariant, 2-forms on $\mathfrak{g}$. and we prove a cohomological $J-$decomposition theorem following \cite{DLZ}: $H^2(\mathfrak{g})=H^+_J(\mathfrak{g})\oplus H^-_J(\mathfrak{g})$. We discover that tameness of $J$ can be characterized in terms of the dimension of $H^{\pm}_J(\mathfrak{g})$, just as in the complex surface case. We also describe the tamed and compatible symplectic cones respectively. Finally, two applications to homogeneous $J$ on 4-manifolds are obtained.

math.SG↗
arXiv2011

Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

Given a Lagrangian sphere in a symplectic 4-manifold $(M, ω)$ with $b^+=1$, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension $κ$ of $(M, ω)$ is $-\infty$, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is not characteristic and ternary, we show that homologous Lagrangian spheres are smoothly isotopic, and when the Euler number is less than 8, we generalize Hind and Evans' Hamiltonian uniqueness in the monotone case. On the existence side, when $κ=-\infty$, we give a characterization of classes represented by Lagrangian spheres, which enables us to describe the non-Torelli part of the symplectic mapping class group.

math.SG↗
arXiv2011

Luttinger surgery and Kodaira dimension

In this note we show that the Lagrangian Luttinger surgery preserves the symplectic Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.

math.GT↗
arXiv2011

Symplectic forms and cohomology decomposition of almost complex 4-manifolds

For any compact almost complex manifold $(M,J)$, the last two authors defined two subgroups $H_J^+(M)$, $H_J^-(M)$ of the degree 2 real de Rham cohomology group $H^2(M, \mathbb{R})$ in arXiv:0708.2520. These are the sets of cohomology classes which can be represented by $J$-invariant, respectively, $J$-anti-invariant real $2-$forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of $H^2(M, \mathbb{R})$. This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given.

math.SG↗
arXiv2011

On the J-anti-invariant cohomology of almost complex 4-manifolds

For a compact almost complex 4-manifold $(M,J)$, we study the subgroups $H^{\pm}_J$ of $H^2(M, \mathbb{R})$ consisting of cohomology classes representable by $J$-invariant, respectively, $J$-anti-invariant 2-forms. If $b^+ =1$, we show that for generic almost complex structures on $M$, the subgroup $H^-_J$ is trivial. Computations of the subgroups and their dimensions $h^{\pm}_J$ are obtained for almost complex structures related to integrable ones. We also prove semi-continuity properties for $h^{\pm}_J$.

math.SG↗
arXiv2009

Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds

We introduce certain homology and cohomology subgroups for any almost complex structure and study their pureness, fullness and duality properties. Motivated by a question of Donaldson, we use these groups to relate J-tamed symplectic cones and J-compatible symplectic cones over a large class of almost complex manifolds, including all Kahler manifolds, almost Kahler 4-manifolds and complex surfaces.

math.SG↗
arXiv2009

Symplectic Birational Geometry

This is a survey on symplectic birational geometry. In arbitrary dimension, this subject is centered around the notion of uniruledness. In low dimensions, we will also discuss Kodaira dimension and minimality.

math.SG↗
arXiv2009

Additivity and Relative Kodaira Dimensions

In this paper we introduce the notion of relative Kodaira dimension for a symplectic 4-manifold M with a possibly disconnected embedded symplectic surface F. One notable feature is that the sphere components of F have to be discarded, which resembles the definition of Thurston norm of 3-manifolds. To define it we need to establish several homological properties of embedded symplectic surfaces. We also formulate the notion of relative minimal model, and prove the existence and the somewhat surprising uniqueness.

math.SG↗
arXiv2008

Existence of Symplectic Surfaces

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be embedded if 2n is at least 6. We also analyze the additional conditions under which embedded symplectic representatives exist in dimension 4.

math.SG↗
arXiv2008

The Relative Symplectic Cone and T^2-Fibrations

In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T^2-fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kaehler surfaces raised by the second author.

math.SG↗
arXiv2007

Birational cobordism invariance of uniruled symplectic manifolds

A symplectic manifold $(M,ω)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternberg's notion of cobordism as the symplectic analogue of the birational equivalence.

math.SG↗
arXiv2007

Uniruled symplectic divisors

This is a paper devoted to the symplectic birational geometry program where many basic notions are defined in terms of genus 0 GW invariants. We show that the existence of a positive uniruled symplectic divisor often implies that the ambient manifold has a nonzero uniruled genus 0 GW invariant, hence is uniruled as well. This confirms a part of the dichotomy on uniruled symplectic divisors. In addition, it gives a rather general construction of uniruled symplectic manifolds, generalizing some beautiful results of McDuff.

math.SG↗
arXiv2006

Symplectic forms and surfaces of negative square

We introduce an analogue of the inflation technique of Lalonde-McDuff, allowing us to obtain new symplectic forms from symplectic surfaces of negative self-intersection in symplectic four-manifolds. We consider the implications of this construction for the symplectic cones of Kähler surfaces, proving along the way a result which can be used to simplify the intersections of distinct pseudoholomorphic curves via a perturbation.

math.SG↗
arXiv2005

Geography of symplectic 4-manifolds with Kodaira dimension one

The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the conclusion of the Hard Lefschetz Theorem, which is measured by a nonnegative integer called the degeneracy. In this paper we include the degeneracy as an extra parameter in the geography problem and show how to fill out the geography of symplectic 4-manifolds with Kodaira dimension 1 for all admissible triples.

math.SG↗