Hiroshi Nakahara
We find calibrated submanifolds in neck manifolds. Particularly, we obtain a calibrated submanifold in the Lagrangian self-expander constructed by Joyce, Lee and Tsui.
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We review some results concerning the deformations of calibrated minimal submanifolds which occur in Riemannian manifolds with special holonomy. The calibrated submanifolds are assumed compact with a non-empty boundary which is constrained to move in a particular fixed submanifold. The results extend McLean's deformation theory previously developed for closed compact submanifolds.
Calibrated submanifolds with flat normal bundles
该研究证明欧氏空间中满足常系数微分形式校准且法丛平坦的子流形为平面，还推广到配备平行校准的黎曼流形中，满足形状算子交换条件的校准子流形是全测地的。
We construct calibrated submanifolds of R^7 and R^8 by viewing them as total spaces of vector bundles and taking appropriate sub-bundles which are naturally defined using certain surfaces in R^4. We construct examples of associative and coassociative submanifolds of R^7 and of Cayley submanifolds of R^8. This construction is a generalization of the Harvey-Lawson bundle construction of special Lagrangian submanifolds of R^{2n}.
Butscher, D. Lee, Y. Lee, and Joyce constructed a special Lagrangian submanifold by gluing a Lawlor neck into a transverse intersection point of two special Lagrangian submanifolds. We prove a uniqueness theorem for the gluing of flat special Lagrangian tori of real dimension 3 in a flat complex torus of complex dimension 3.
`Gluing' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing technique. We have existence theorems in that sense, but there seems to have been no uniqueness theory for higher-dimensional ones such as special Lagrangian submanifolds, which we discuss in the present paper.
Calibrated submanifolds, adiabatic limit, and gradient graphs
该研究针对半平SYZ纤维化中特殊拉格朗日量的紧性问题，给出绝热极限出现梯度图的判据，是Donaldson-Scaduto方案的部分逆命题。
We provide an introduction to the theory of calibrated submanifolds through the key examples related with special holonomy. We focus on calibrated geometry in Calabi-Yau, G$_2$ and Spin(7) manifolds, and describe fundamental results and techniques in the field.
For an element $Ψ$ in the graded vector space $Ω^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $Ψ$-submanifold is defined as a submanifold $N$ of $M$ such that $Ψ_{|N} \in Ω^*(N, TN)$. The class of $Ψ$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space $Ω^*(M, TM)$ carries a natural graded Lie algebra structure, given by the Frölicher-Nijenhuis bracket $[-,- ]^{FN}$. When $Ψ$ is an odd degree element with $[ Ψ, Ψ]^{FN} =0$, we associate to a $Ψ$-submanifold $N$ a strongly homotopy Li…
The principal theory of this paper comprises a technique for constructing associative, coassociative and Cayley submanifolds of Euclidean space with symmetries, using first-order ordinary differential equations. Explicit examples of U(1)-invariant associative cones in R^7 and SU(2)-invariant Cayley 4-folds in R^8 are then produced using this method. Further examples of associative 3-folds are presented, which are ruled, and other systems of differential equations defining calibrated submanifolds in R^7 and R^8 are given.
Bernstein Theorems for Calibrated Submanifolds in $\mathbb{R}^7$ and $\mathbb{R}^8$
本文研究$\u211d^7$中余结合子流形对应的$\u211d^4$到$\u211d^3$光滑映射的Bernstein问题，给出由斜率第二初等对称多项式表述的仿射性条件，并得到$\u211d^8$中Cayley子流形的对应定理。
This paper is a continuation of math.DG/0408005. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S^n by looking at the conormal bundle of appropriate submanifolds of S^n. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey-Lawson for submanifolds in R^n in their pioneering paper. We also construct calibrated submanifolds in complete metrics with special holonomy G_2 and Spin(7) discovered by Bryant and Salamon on the total spaces of appropriate bun…
Some results on calibrated submanifolds in Euclidean space of cohomogeneity one and two
本文研究了在欧几里得空间中具有共效性的子流形，通过构造特殊Lagrangian子流形并证明其刚性结果。
Extrinsic geometry of calibrated submanifolds
本文针对稳定子群可迁作用的校准α提出李理论条件“顺应性”，证明其适用于多类经典校准并完成刻画，同时刻画了平行顺应校准下黎曼浸入的外蕴几何约束，推广了超极小曲面概念。
In this paper we will investigate torus actions on complete manifolds with calibrations. For Calabi-Yau manifolds M^2n with a Hamiltonian structure-preserving k-torus action we show that any symplectic reduction has a natural holomorphic volume form. Moreover Special Lagrangian (SLag) submanifolds of the reduction lift to SLag submanifolds of M, invariant under the torus action. If k=n-1 and the first cohomology of M is trivial, then we prove that M is a fibration with generic fiber being a SLag submanifold. As an application we will see that crepant resolutions of singularities of a finite Ab…
A variational characterization of calibrated submanifolds
该研究通过特定方式改变流形上的校准构造外围度量的特殊变分，证明紧定向嵌入子流形被校准当且仅当外围度量是体积泛函的临界点，推广了殆凯勒情形的已有结果，并应用于Smith映射的变分刻画。
Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of c…
Anisotropic calibrations, Fueter maps and mirror symmetry
本文研究了各向异性校准、绝热极限及其与镜像对称的关系，通过定义1参数形式α_ε并研究其绝热极限，证明了其在广义意义下的校准性质，并展示了绝热校准子流形在变分/偏微分方程理论中的各向异性极小性。
We show the total space of the canonical line bundle $\mathbb{L}$ of a Kahler-Einstein manifold $X^n$ supports integrable $SU(n+1)$ structures, or Calabi-Yau structures. The canonical real line bundle $L \subset \mathbb{L}$ over a minimal Lagrangian submanifold $M \subset X$ is calibrated in this setting and hence can be considered as the special Lagrangian lift of $M$. For the integrable $G_2$ and $Spin(7)$ structures on spin bundles and bundles of anti-self-dual 2-forms on self-dual Einstein 4-manifolds constructed by Bryant and Salamon, minimal surfaces with vanishing complex quartic form (…
Let H be the hyperbolic space of dimension n+1. A geodesic foliation of H is given by a smooth unit vector field on H all of whose integral curves are geodesics. Each geodesic foliation of H determines an n-dimensional submanifold M of the 2n-dimensional manifold L of all the oriented geodesics of H (up to orientation preserving reparametrizations). The space L has a canonical split semi-Riemannian metric induced by the Killing form of the isometry group of H. Using a split special Lagrangian calibration, we study the volume maximization problem for a certain class of geometrically distinguish…
Given a parallel calibration $ϕ\in Ω^p(M)$ on a Riemannian manifold $M$, I prove that the $ϕ$--critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the $ϕ$--critical submanifolds are precisely the integral manifolds of a $\mathscr{C}^\infty(M)$--linear subspace $\sP \subset Ω^p(M)$. In particular, the calibrated submanifolds are necessarily integral submanifolds of the system. (Examples of parallel calibrations include the special Lagrangian calibration on Calabi-Yau manifolds, (co)associative calibrations on $G_2$--manifolds, and the Cayley calibration …
We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean cu…
In this paper we investigate the geometry of Calibrated submanifolds and study relations between their moduli-space and geometry of the ambient manifold. In particular for a Calabi-Yau manifold we define Special Lagrangian submanifolds for any Kahler metric on it. We show that for a choice of Kahler metric the Borcea-Voisin threefold has a fibration structure with generic fiber being a Special Lagrangian torus. Moreover we construct a mirror to this fibration. Also for any closed G_2 form on a 7-manifold we study coassociative submanifolds and we show that one example of a G_2 manifold constru…
For Hitchin's generalised geometries we introduce and analyse the concept of a structured submanifold which encapsulates the classical notion of a calibrated submanifold. Under a suitable integrability condition on the ambient geometry, these generalised calibrated cycles minimise a functional occurring as D-brane energy in type II string theories, involving both so-called NS-NS- and R-R-fields. Further, we investigate the behaviour of calibrated cycles under T-duality and construct non-trivial examples.
A nearly parallel $G_{2}$-manifold $Y$ is a Riemannian 7-manifold whose cone $C(Y) = \mathbb{R}_{>0} \times Y$ has the holonomy group contained in ${\rm Spin(7)}$. In other words, it is a spin 7-manifold with a real Killing spinor. We have a special class of calibrated submanifolds called Cayley submanifolds in $C(Y)$. An associative submanifold in $Y$ is a minimal 3-submanifold whose cone is Cayley. We study its deformations, namely, Cayley cone deformations, explicitly when it is homogeneous in the 7-sphere $S^{7}$.
Coassociative submanifolds are 4-dimensional calibrated submanifolds in $G_{2}$-manifolds. In this paper, we construct explicit examples of coassociative submanifolds in $Λ^{2}_{-} S^{4}$, which is the complete $G_{2}$-manifold constructed by Bryant and Salamon. Classifying the Lie groups which have 3- or 4-dimensional orbits, we show that the only homogeneous coassociative submanifold is the zero section of $Λ^{2}_{-} S^{4}$ up to the automorphisms and construct many cohomogeneity one examples explicitly. In particular, we obtain examples of non-compact coassociative submanifolds with conical…
We revisit McLean's second variation formulas for calibrated submanifolds in exceptional geometries, and correct his formulas concerning associative submanifolds and Cayley submanifolds, using a unified treatment based on the (relative) calibration method and Harvey-Lawson's identities.
By a theorem of Mclean, the deformation space of an associative submanifold Y of an integrable G_2 manifold (M,ϕ) can be identified with the kernel of a Dirac operator D:Ω^{0}(ν) -->Ω^{0}(ν) on the normal bundle νof Y. Here, we generalize this to the non-integrable case, and also show that the deformation space becomes smooth after perturbing it by natural parameters, which corresponds to moving Y through `pseudo-associative' submanifolds. Infinitesimally, this corresponds to twisting the Dirac operator D --> D_{A} with connections A of ν. Furthermore, the normal bundles of the associative sub…
On a Riemannian manifold $\bar{M}^{m+n}$ with an $(m+1)$-calibration $Ω$, we prove that an $m$-submanifold $M$ with constant mean curvature $H$ and calibrated extended tangent space $\mathbb{R}H\oplus TM$ is a critical point of the area functional for variations that preserve the enclosed $Ω$-volume. This recovers the case described by Barbosa, do Carmo and Eschenburg, when $n=1$ and $Ω$ is the volume element of $\bar{M}$. To the second variation we associate an $Ω$-Jacobi operator and define $Ω$-stablility. Under natural conditions, we prove that the Euclidean $m$-spheres are the unique $Ω$-s…
The aim of this paper is to study variational properties for $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials. Examples of submanifolds of this kind incuding soliton solutions for Lagrangian mean curvature flow (LMCF). We derive second variation formula for $f$-minimal Lagrangians as a generalization of Chen and Oh's formula for minimal Lagrangians. As a corollary, we obtain stability of expanding and translating solitons for LMCF. We also define calibrated submanifolds with respect to $f$-volume in gradient steady Kähler--Ricci solitons as generalization…
Smooth maps minimizing the energy and the calibrated geometry
该研究将校准子流形概念推广至光滑映射，验证了微分几何中相关实例的适配性，同时给出黎曼流形同伦类光滑映射能量下界，探究特殊和乐群流形上恒等映射极小化的能量泛函。
The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on exceptional holonomy, in two parts. Part I introduces the exceptional holonomy groups, and explains constructions for compact 7- and 8-manifolds with holonomy G2 and Spin(7). The simplest such constructions work by using techniques from complex geometry and Calabi-Yau analysis to resolve the singularities of a torus orbifold T^7/G or T^8/G, for G a finite group preserving a flat G2 or Spin(7)-structure on T^7 or T^8. Ther…
Using the Cartan-Kahler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G_2-manifold as an associative submanifold. Second, the interior of a smooth, compact 4-manifold K, whose double has a trivial bundle of self-dual 2-forms, may be isometrically embedded into a Spin(7)-manifold as a Cayley submanifold. Along the way, we also show that Bochner's Theorem on real analytic approximation of smooth differential forms, can be obtained using real algebraic tools developed by A…
In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold is Lagrangian and almost calibrated by ReΩin a Calabi-Yau n-fold (M,Ω), and T>0 is the first blow-up time of the mean curvature flow, then the tangent cone of the mean curvature flow at a singular point (X,T) is a stationary Lagrangian integer multiplicity current in R\sup 2n with volume density greater than one at X. When n=2, the tangent cone consists of a finite union of more than one 2-planes in R\sup 4 which are complex in a complex structure on R\su…
In this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy many of their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. This paper investigates, in depth, questions of: pseudo-convexity and cores, positive phi-currents, Duval-Sibony Duality, and boundari…
Calibrated Geometry in Hyperkahler Cones, 3-Sasakian Manifolds, and Twistor Spaces
本文系统研究超凯勒锥、3-Sasakian流形和扭量空间中的校准几何，通过典范结构揭示半校准族，并推广了复拉格朗日锥刻画及Storm定理。
In this paper, we show that the calibrated method can also be used to detect indefinite minimal Lagrangian submanifolds in $C_k^m$. We introduce the notion of indefinite special Lagrangian submanifolds in $C_k^m$ and generalize the well-known work of Harvey-Lawson to the indefinite case.
Given $(\bar{M},Ω)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $\cos θ$ is bounded away from zero, where $θ$ is the $Ω$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricc^M\geq 0$ we may replace the boundedness condition on $\cos θ$ by $\cos θ\geq Cr^{-β}$, when $r\to +\infty$, where $ 0\leqβ<1 $ and $C > 0$ are constants and $r$ is the distance function to a point in $M$. Our proof is surprisingly …
The geometry of submanifolds is intimately related to the theory of functions and vector bundles. It has been of fundamental importance to find out how those two objects interact in many geometric and physical problems. A typical example of this relation is that the Picard group of line bundles on an algebraic manifold is isomorphic to the group of divisors, which is generated by holomorphic hypersurfaces modulo linear equivalence. A similar correspondence can be made between the K-group of sheaves and the Chow ring of holomorphic cycles. There are two more very recent examples of such a relat…
Special Lagrangian submanifolds are submanifolds of a Calabi-Yau manifold calibrated by the real part of the holomorphic volume form. In this paper we use elliptic theory for edge-degenerate differential operators on singular manifolds to study the moduli space of deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.
Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry
本文研究具有圆对称性的hyperkähler 4-流形中的拉格朗日平均曲率流，证明其可穿过有限时间奇点延续并收敛至special Lagrangians链，验证了Joyce猜想并指出奇点为颈缩。
We extend the "bundle constructions" of calibrated submanifolds, due to Harvey--Lawson in the special Lagrangian case, and to Ionel--Karigiannis--Min-Oo in the cases of exceptional calibrations, by "twisting" the bundles by a special (harmonic, holomorphic, parallel) section of a complementary bundle. The existence of such deformations shows that the moduli space of calibrated deformations of these "calibrated subbundles" includes deformations which destroy the linear structure of the fibre.
We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kähler case, by its analogues. To this end we explore the geometry of totally real submanifolds, defining (i) a new geometric flow in terms of the ambient canonical bundle, (ii) a modified volume functional which takes into account the totally real condition. We discuss short-time existence for our flow and show it couples …
Deformations of calibrated subbundles in noncompact manifolds of special holonomy via twisting by special sections
本文研究了Calabi-Yau、G2和Spin(7)非紧流形中校准子丛的形变，通过互补丛截面对子丛进行扭曲，推导了保持校准的条件，并发现特殊拉格朗日构造较欧氏空间更为刚性。
A special class of $k$-harmonic maps inducing calibrated fibrations
本文研究了两种特殊的k-调和映射，探讨了其与校准几何的关系，并提出了新的PDE问题及未来研究方向。
We introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. In particular, the notion of pseudo-convexity for a calibrated manifold (X,ϕ) is introduced and studied. Analogues of totally real submanifolds are also int…
In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. We give self-contained proofs here. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. For the sake of completeness we decided to collect them here in a self-contained way to be easily accessible for future usage in calibrated geometry. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of G_2 manifolds with special properties, which…
We study massless deformations of generalized calibrated cycles, which describe, in the language of generalized complex geometry, supersymmetric D-branes in N=1 supersymmetric compactifications with fluxes. We find that the deformations are classified by the first cohomology group of a Lie algebroid canonically associated to the generalized calibrated cycle, seen as a generalized complex submanifold with respect to the integrable generalized complex structure of the bulk. We provide examples in the SU(3) structure case and in a `genuine' generalized complex structure case. We discuss cases of …
Suppose $M_{1}$ and $M_{2}$ are two special Lagrangian submanifolds of $\Rtn$ with boundary that intersect transversally at one point $p$. The set $M_{1} \cup M_{2}$ is a singular special Lagrangian variety with an isolated singularity at the point of intersection. Suppose further that the tangent planes at the intersection satisfy an angle condition (which always holds in dimension $n=3$). Then, $M_{1} \cup M_{2}$ is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds $M_α$ with boundary that converges to $M_{1} \cup M_{2}$ in a suitable topology. T…
On $G_2$ manifolds with cohomogeneity two symmetry
本文研究了具有余齐次二对称性的 $G_2$ 流形，给出了局部刻画及余齐次一校准子流形的几何与奇性分析，并在特定流形中发现了新的完备 $\mathbb{T}^2$-不变结合子流形族。
Entire conformal curves are affine or have super-Euclidean energy growth
该研究证明高维整共形曲线仅分为仿射曲线和超欧几里得能量增长两类，推导了相关刚性结论与应用，并给出了子流形共形等价的相关结果。
A rescaling principle for quasiregular curves with applications to hyperbolicity
研究拟正则曲线到校准流形的重标度原理，通过引入适应设定的布罗迪双曲性等方法，证明了相关等价性，量化了局部连续性模，回答了问题并构造新例子给出布罗迪双曲性障碍。
In this paper, we prove various results on the topology of the Grassmannian of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We give self-contained proofs. These spaces come up when studying submanifolds of manifolds with calibrated geometries. We collect these results here for the sake of completeness. As applications of our algebraic topological study we present some results on special Lagrangian-free embeddings of surfaces and 3-manifolds into the Euclidean 4 and 6-space.
We find Weitzenböck formula for the Fueter-Dirac operator which controls the infinitesimal deformations of an associative submanifold in a $7$--manifold with a $G_2$--structure. We establish a vanishing theorem to conclude rigidity under some positivity assumptions on curvature, which are particularly mild in the nearly parallel case. As applications, we give a different proof of rigidity for one of Lotay's associatives in the round $7$-sphere from those given by Kawai. We also provide simpler proofs of previous results by Gayet for the Bryant-Salamon metric. Finally, we obtain an original exa…
This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject, and is designed to be fairly self-contained. It is based on lecture courses given at Nordfjordeid, Norway and MSRI, Berkeley in June and July 2001. We introduce Calabi-Yau m-folds via holonomy groups, Kahler g…
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