In this paper, we show that there exists no equifocal submanifold with non-flat section in four irreducible simply connected symmetric spaces of compact type and rank two. Also, we show a fact for the sections of equifocal submanifolds with non-flat
section in other irreducible simply connected symmetric spaces of compact type and rank two.
Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean
curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres.
In this paper we are interested in defining affine structures on discrete quadrangular surfaces of the affine three-space. We introduce, in a constructive way, two classes of such surfaces, called respectively indefinite and definite surfaces. The un
derlying meshes for indefinite surfaces are asymptotic nets satisfying a non-degeneracy condition, while the underlying meshes for definite surfaces are non-degenerate conjugate nets satisfying a certain natural condition. In both cases we associate to any of these nets several discrete affine invariant quantities: a metric, a normal and a co-normal vector fields, and a mean curvature. Moreover, we derive structural and compatibility equations which are shown to be necessary and sufficient conditions for the existence of a discrete quadrangular surface with a given affine structure.
Let X be an n-dimensional Calabi-Yau with ordinary double points, where n is odd. Friedman showed that for n=3 the existence of a smoothing of X implies a specific type of relation between homology classes on a resolution of X. (The converse is also
true, due to work of Friedman, Kawamata and Tian.) We sketch a more topological proof of this result, and then extend it to higher dimensions. For n>3 the Yukawa product on the middle dimensional (co)homology plays an unexpected role. We also discuss a converse, proving it for nodal Calabi-Yau hypersurfaces in projective space.
A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and
sphere. We show that the mean curvature flow preserves the isoparametric condition, develops singularities in finite time, and converges in finite time to a smooth submanifold of lower dimension. We also give a precise description of the collapsing.
We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, omega, tau)$ with nonempty and compact real part $L={rm Fix}(tau)$. For given $Lambdain (0, +infty]$ and $minNcup{0}$ we show the equivalence of
the following two claims: (i) $sharp(Lcapphi^H_1(L))ge m$ for any Hamiltonian function $Hin C_0^infty([0, 1]times M)$ with Hofers norm $|H|<Lambda$; (ii) $sharp {cal P}(H,tau)ge m$ for every $Hin C^infty_0(R/Ztimes M)$ satisfying $H(t,x)=H(-t,tau(x));forall (t,x)inmathbb{R}times M$ and with Hofers norm $|H|<2Lambda$, where ${cal P}(H, tau)$ is the set of all $1$-periodic solutions of $dot{x}(t)=X_{H}(t,x(t))$ satisfying $x(-t)=tau(x(t));forall tinR$ (which are also called brake orbits sometimes). Suppose that $(M, omega)$ is geometrical bounded for some $Jin{cal J}(M,omega)$ with $tau^ast J=-J$ and has a rationality index $r_omega>0$ or $r_omega=+infty$. Using Hofers method we prove that if the Hamiltonian $H$ in (ii) above has Hofers norm $|H|<r_omega$ then $sharp(Lcapphi^H_1(L))gesharp {cal P}_0(H,tau)ge {rm Cuplength}_{F}(L)$ for $F=Z_2$, and further for $F=Z$ if $L$ is orientable, where ${cal P}_0(H,tau)$ consists of all contractible solutions in ${cal P}(H,tau)$.
In this expository note, we study the second variation of Perelmans entropy on the space of Kahler metrics at a Kahler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelmans entropy is stable along the
Kahler-Ricci flow. The Chinese version of this note has appeared in a volume in honor of professor K.C.Chang (Scientia Sinica Math., 46 (2016), 685-696).
In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(Gamma) of a compact Lie
group $Gamma$ to the complex K-theory of the classifying space $BGamma$. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlssons deformation $K$--theory spectrum $K (Gamma)$ (the homotopy-theoretical analogue of $R(Gamma)$). Our main theorem provides an isomorphism in homotopy $K_*(pi_1 Sigma)isom K^{-*}(Sigma)$ for all compact, aspherical surfaces $Sigma$ and all $*>0$. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.
We revisit Atiyah and Botts study of Morse theory for the Yang-Mills functional over a Riemann surface, and establish new formulas for the minimum codimension of a (non-semi-stable) stratum. These results yield the exact connectivity of the natural m
ap (C_{min} E)//G(E) --> Map^E (M, BU(n)) from the homotopy orbits of the space of central Yang-Mills connections to the classifying space of the gauge group G(E). All of these results carry over to non-orientable surfaces via Ho and Lius non-orientable Yang-Mills theory. A somewhat less detailed version of this paper (titled On the Yang-Mills stratification for surfaces) will appear in the Proceedings of the AMS.
We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy typ
e of the infinite symmetric product of M^g, generalizing a well-known fact for the torus. Over a non-orientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori, whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surfaces, and show a close analogy with the Quillen-Lichtenbaum conjectures in algebraic K-theory. The proofs utilize Tyler Lawsons work in deformation K-theory, and rely heavily on Yang-Mills theory and gauge theory.
