Some families of special Lagrangian tori

 We give an explicit proof of the local version of Bryant's result [1], stating that any 3-dimensional real-analytic Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We then refine the theorem proving that a certain class of real-analytic one-parameter families of metrics on a 3-torus can be isometrically embedded in a Calabi-Yau manifold as a one-parameter family of special Lagrangian submanifolds. Two applications of these results show how the geometry of the moduli space of 3-dimesional special Lagrangian submanifolds differs considerably from the 2-dimensional one. First of all, applying Bryant's theorem and a construction due to Calabi we show that nearby elements of the local moduli space of a special Lagrangian 3-torus can intersect themselves. Secondly, we use our examples of one-parameter families to show that in dimension three (and higher) the moduli space of special Lagrangian tori is not, in general, special Lagrangian in the sense of Hitchin [13]. Received: 18 December 2001 / Revised version: 31 January 2002 / Published online: 16 October 2002 Mathematics Subject Classification (2000): 53-XX, 53C38     

On families of Lagrangian submanifolds

We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ω _t are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ _t of diffeomorphisms of X such that ω _t =Φ _t ^*(ω₀). If L⊂X is a Lagrangian submanifold for (X,ω₀), L _t =Φ _t ^-1(L) is thus a Lagrangian submanifold for (X,ω _t ). Here we show that if we simply assume that L is compact and ω _t | _L is exact for every t, a family L _t as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi–Yau structure. Received: 29 May 2001/ Revised version: 17 October 2001     

Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities: I. The gluing construction

In this paper we extend our previous results on resolving conically singular Calabi–Yau 3-folds (Chan, Quart. J. Math. 57:151–181, 2006; Quart. J. Math., to appear) to include the desingularizations of special Lagrangian (SL) 3-folds with conical singularities that occur at the same points of the ambient Calabi–Yau. The gluing construction of the SL 3-folds is achieved by applying Joyce’s analytic result (Joyce, Ann. Global. Anal. Geom. 26: 1–58, 2004, Thm. 5.3) on deforming Lagrangian submanifolds to nearby special Lagrangian submanifolds. Our result will in principle be able to construct more examples of compact SL submanifolds in compact Calabi–Yau manifolds. Various explicit examples and applications illustrating the result in this paper can be found in the sequel (Chan, Ann. Global. Anal. Geom., to appear).     

Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities. II. Examples

This article is a sequel to Chan (Ann Glob Anal Geom, to appear) on simultaneous desingularizations of Calabi–Yau and special Lagrangian (SL) 3-folds with conical singularities. In Chan (Ann Glob Anal Geom, to appear) we treated the question of starting with a conically singular Calabi–Yau 3-fold and an SL 3-fold with conical singularities at the same points and deforming both together to get a smooth situation. In this article, we survey the major result from Chan (Ann Glob Anal Geom, to appear) and describe some examples from our earlier articles (Chan Q J Math 57:151–181, 2006, Q J Math, to appear) on Calabi–Yau desingularizations. We then provide many explicit examples of Asymptotically Conical (AC) SL submanifolds in two specific AC Calabi–Yau manifolds. Using the result in Chan (Ann Glob Anal Geom, to appear), we construct smooth examples of compact SL 3-folds in compact Calabi–Yau 3-folds by gluing those AC SL 3-folds into some conically singular SL 3-folds at the singular points.     

Calibrated Embeddings in the Special Lagrangian and Coassociative Cases

Every closed, oriented, real analytic Riemannian3–manifold can be isometrically embedded as a specialLagrangian submanifold of a Calabi–Yau 3–fold, even as thereal locus of an antiholomorphic, isometric involution. Every closed,oriented, real analytic Riemannian 4–manifold whose bundle of self-dual2–forms is trivial can be isometrically embedded as a coassociativesubmanifold in a G₂-manifold, even as the fixed locus of ananti-G₂ involution.These results, when coupledwith McLean's analysis of the moduli spaces of such calibratedsubmanifolds, yield a plentiful supply of examples of compact calibratedsubmanifolds with nontrivial deformation spaces.     

New examples of compact manifolds with holonomy Spin(7)

We find new examples of compact Spin(7)-manifolds using a construction of Joyce (J. Differ. Geom., 53:89–130, 1999; Compact manifolds with special holonomy. Oxford University Press, Oxford, 2000). The essential ingredient in Joyce’s construction is a Calabi–Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi–Yau 4-orbifolds.     

Isometric Actions of Heisenberg Groups on Compact Lorentz Manifolds

We prove results toward classifying compact Lorentz manifolds on which Heisenberg groups act isometrically. We give a general construction, leading to a new example, of codimension-one actions – those for which the dimension of the Heisenberg group is one less than the dimension of the manifold. The main result is a classification of codimension-one actions, under the assumption they are real-analytic.     

DESINGULARIZATIONS OF CALABI-YAU 3-FOLDS WITH CONICAL SINGULARITIES. II. THE OBSTRUCTED CASE

This is the second of two papers studying Calabi–Yau 3-foldswith conical singularities and their desingularizations. Inour first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181]we constructed the desingularization of the conically singularmanifold M₀ by gluing an asymptotically conical (AC) Calabi–Yau3-fold Y into M₀ at the singular point, thus obtaining a 1-parameterfamily of compact, non-singular Calabi–Yau 3-folds M_tfor small t > 0. During the gluing process one may encountera kind of cohomological obstruction to defining a 3-form _t onM_t which interpolates between the 3-form ₀ on M₀ and the scaled3-form t³ _Y on Y if the rate at which the AC Calabi–Yau3-fold Y converges to the Calabi–Yau cone is equal to– 3. The first paper [3] studied the simpler case <–3 where there is no obstruction. This paper extends theresult in the first one by considering a more complicated situtationwhen = –3. Assuming the existence of singular Calabi–Yaumetrics on compact complex 3-folds with ordinary double points,our result in this paper can be applied to repairing such kindsof singularities, which is an analytic version of Friedman'sresult giving necessary and sufficient conditions for smoothingordinary double points.     

On the global theory of projective mappings

We consider the theory of constant rank projective mappings of compact Riemannian manifolds from the global point of view. We study projective immersions and submersions. As an example of the results, letf:(M, g) → (N, g′) be a projective submersion of anm-dimensional Riemannian manifold (M, g) onto an (m−1)-dimensional Riemannian manifold (N, g′). Then (M, g) is locally the Riemannian product of the sheets of two integrable distributions Kerf _* and (Kerf _*)^⊥ whenever (M, g) is one of the two following types: (a) a complete manifold with Ric ≥ 0; (b) a compact oriented manifold with Ric ≤ 0. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 111–118, July, 1995. This work was partially supported by the Russian Foundation for Basic Research grant No. 94-01-0195.     

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions. In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact Riemannian manifolds.     

On monodromies of a degeneration of irreducible symplectic Kähler manifolds

We study the monodromy operators on the Betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension. This implies that the “mildest” singular fiber of a good degeneration with non-trivial monodromy of irreducible symplectic manifolds is quite different from the generic degeneration of abelian varieties or Calabi–Yau manifolds.     

Isolated rational curves on K3-fibered Calabi–Yau threefolds

In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙ ^{n 1}}×ℙ¹. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau variety in projective space. Received: 14 October 1997 / Revised version: 18 January 1998     

A family of Kähler-Einstein manifolds and metric rigidity of Grauert tubes

In this paper we explain how the so-called adapted complex structures can be used to associate to each compact real-analytic Riemannian manifold a family of complete Kähler-Einstein metrics and show that already one element of this family uniquely determines the original manifold. The underlying manifolds of these metrics are open disc bundles in the tangent bundle of the original Riemannian manifold.

     

A LOWER BOUND FOR FIRST EIGENVALUE WITH MIXED BOUNDARY CONDITIOIN

Let M be an n-dimensional compact Riemannian manifold with or without boundary,and its Ricci curvature RicM≥n- 1. The paper obtains an inequality for the first eigenvalue η1 of M with mixed boundary condition, which is a generalization of the results of Lichnerowicz,Reilly, Escobar and Xia. It is also proved that η1≥ n for certain n-dimensional compact Riemannian manifolds with boundary,which is an extension of the work of Cheng,Li and Yau.     

Conformally Einstein products and nearly Kähler manifolds

In the first part of this note we study compact Riemannian manifolds (M, g) whose Riemannian product with is conformally Einstein. We then consider 6-dimensional almost Hermitian manifolds of type W ₁ + W ₄ in the Gray–Hervella classification admitting a parallel vector field and show that (under some mild assumption) they are obtained as Riemannian cylinders over compact Sasaki–Einstein 5-dimensional manifolds.      

Systolic Inequalities and Minimal Hypersurfaces

We give a short proof of the systolic inequality for the n-dimensional torus. The proof uses minimal hypersurfaces. It is based on the Schoen–Yau proof that an n-dimensional torus admits no metric of positive scalar curvature.     

Examples of Calabi–Yau Threefolds as Covers of Almost-Fano Threefolds

We give a method for producing examples of Calabi–Yau threefolds as covers of degree d ≤ 8 of almost-Fano threefolds, computing explicitely their Euler– Poincaré characteristic. Such a method generalizes the well-knownclassical construction of Calabi–Yau threefolds as double covers of the projective space branched along octic surfaces.     

Enumerative meaning of mirror maps for toric Calabi–Yau manifolds

We prove that the inverse of a mirror map for a toric Calabi–Yau manifold of the form _KY$K_Y$, where Y$Y$ is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov–Witten invariants defined by Fukaya–Oh–Ohta–Ono (2010)  [15]. Such a relation between mirror maps and disk counting invariants was first conjectured by Gross and Siebert (2011)  [24, Conjecture 0.2 and Remark 5.1] as part of their program, and was later formulated in terms of Fukaya–Oh–Ohta–Ono’s invariants in the toric Calabi–Yau case in Chan et al. (2012)  [8, Conjecture 1.1].     

Nodal volumes for eigenfunctions of analytic regular elliptic problems

We establish sharp upper bounds on the (n−1)-dimensional Hausdorff measure of the zero (nodal) sets and on the maximal order of vanishing corresponding to eigenfunctions of a regular elliptic problem on a bounded domain Ω ⊆ ℝ ⁿ with real-analytic boundary. The elliptic operator may be of an arbitrary even order, and its coefficients are assumed to be real-analytic. This extends a result of Donnelly and Fefferman ([DF1], [DF3]) concerning upper bounds for nodal volumes of eigenfunctions corresponding to the Laplacian on compact Riemannian manifolds with boundary.