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We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of some general Sturm-type theorems.
We give a classification of quadratic harmonic morphisms between Euclidean spaces (Theorem 2.4) after proving a Rank Lemma. We also find a correspondence between umbilical (Definition 2.7) quadratic harmonic morphisms and Clifford systems. In the case $ {\Bbb R}^{4}\longrightarrow {\Bbb R}^{3} $, we determine all quadratic harmonic morphisms and show that, up to a constant factor, they are all bi-equivalent (Definition 3.2) to the well-known Hopf construction map and induce harmonic morphisms bi-equivalent to the Hopf fibration ${\Bbb S}^{3} \longrightarrow {\Bbb S}^{2}$.
We introduce the complete lifts of maps between (real and complex) Euclidean spaces and study their properties concerning holomorphicity, harmonicity and horizontal weakly conformality. As applications, we are able to use this concept to characterize holomorphic maps $ϕ:{\Bbb C}^{m}\supset U\longrightarrow {\Bbb C}^{n}$ (Proposition 2.3) and to construct many new examples of harmonic morphisms (Theorem 3.3). Finally we show that the complete lift of the quaternion product followed by the complex product is a simple and explicit example of a harmonic morphism which does not arise (see Definition 4.8 in \cite{BaiWoo95}) from any K{ä}hler structure.
In a complete Riemannian manifold $(M, g)$ if the hessian of a real valued function satisfies some suitable conditions then it restricts the geometry of $(M, g)$. In this paper we characterize all compact rank-1 symmetric spaces, as those Riemannian manifolds $(M, g)$ admitting a real valued function $u$ such that the hessian of $u$ has atmost two eigenvalues $-u$ and $-{{u+1}\over 2}$, under some mild hypothesis on $(M, g)$. This generalises a well known result of Obata which characterizes all round spheres.
We use rudiments of the Seiberg-Witten gluing theory for trivial circle bundles over a Riemann surface to relate de Seiberg-Witten basic classes of two $4$-manifolds containing Riemann surfaces of the same genus and self-intersection zero with those of the $4$-manifold resulting as a connected sum along the surface. We study examples in which this is enough to describe completely the basic classes.
Antonio Ros gave a lower bound for the first eigenvalue $λ_1$ of $Δ$ of a $P$-manifold $(M, g)$ in terms of the lower bound on the Ricci curvature $Ric_M$ and asked what happened when this lower bound was achieved. In this paper we look in to this question and show that there are strong implications on the geometry and topology of the underlying manifold. In particular we show that in case of spheres or real projective spaces we have isometry with the standard metric. In other cases, with some additional hypothesis, we again show isometry with standard models.
We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making `analytic' connected sums. In particular, we can easily construct complete metrics of constant positive scalar curvature on the complement of certain configurations of an even number of points on the sphere, which is a special case of Schoen's \cite{S1} well-known, difficult construction. Applications of this construction produces metrics with prescribed asymptotics. In particular, we produce metrics with cylindrical ends, the simplest type of asymptotic behaviour. Solutions on the complement of an infinite number of points are also constructed by an iteration of our construction.
The spinor representation is developed and used to investigate minimal surfaces in ${\bfR}^3$ with embedded planar ends. The moduli spaces of planar-ended minimal spheres and real projective planes are determined, and new families of minimal tori and Klein bottles are given. These surfaces compactify in $S^3$ to yield surfaces critical for the Möbius invariant squared mean curvature functional $W$. On the other hand, all $W\!$-critical spheres and real projective planes arise this way. Thus we determine at the same time the moduli spaces of $W\!$-critical spheres and real projective planes via the spinor representation.
We give results on the following questions about a topologically tame hyperbolic 3-manifold M : 1. Does M have nonzero square-integrable harmonic 1-forms? 2. Does zero lie in the spectrum of the Laplacian acting on (1-forms on M)/Ker(d)?
We introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like Yang-Mills and string theory and which generalise finite dimensional Riemannian principal fibre bundles induced by an isometric action. We show that the orbits of regularisable bundles have well defined, both heat-kernel and zeta function regularised volumes. We introduce two notions of minimality (which extend the finite dimensional one) for these orbits, using both heat-kernel and zeta function regularisation methods and show they coincide. For each of these notions, we give an infinite dimensional version of Hsiang's theorem which extends the finite dimensional case, interpreting minimal orbits as orbits with extremal (regularised) volume.
A classical result in differential geometry due to Lichnerowicz [8] is concerned with the decomposition of the square of Dirac operators defined by Clifford connections on a Clifford module ${\cal E}$\ over a Riemannian manifold $M$. Recently, this formula has been generalized to arbitrary Dirac operators [2]. In this paper we prove a supersymmetric version of the generalized Lichnerowicz formula, motivated by the fact that there is a one-to-one correspondence between Clifford superconnections and Dirac operators. We extend this result to obtain a simple formula for the supercurvature of a generalized Bismut superconnection. This might be seen as a first step to prove the local index theorem also for families of arbitrary Dirac operators.
The negative case of the Singular Yamabe Problem concerns the existence and behavior of complete metrics with constant negative scalar curvature on the complement of a closed set in a compact Riemannian manifold which are conformally equivalent to a smooth metric on this compact manifold. When the closed set is a smooth submanifold, it is known by the results of Loewner-Nirenberg and Aviles-McOwen that there exists such a complete metric if and only if $d > (n-2)/2$, and in general the Hausdorff dimension of the set must be at least $(n-2)/2$. In this paper, we show that the existence of such a complete conformal metric with constant negative scalar curvature depends on the tangent structure of the closed set. Specifically, provided the set has a nice tangent cone at a point, we show that when the dimension of this tangent cone is less than $(n-2)/2$ there can not exist such a negative Singular Yamabe metric.
In this paper we survey a number of recent results concerning the existence and moduli spaces of solutions of various geometric problems on noncompact manifolds. The three problems which we discuss in detail are: I. Complete properly immersed minimal surfaces in $\RR^3$ with finite total curvature. II. Complete embedded surfaces of constant mean curvature in $\RR^3$ with finite topology. III. Complete conformal metrics of constant positive scalar curvature on $M^n \setminus Λ$, where $M^n$ is a compact Riemannian manifold, $n\geq3$ and $\Lam \subset M$ is closed. The existence results we discuss for each of these problems are ones whereby known solutions (sometimes satisfying certain nondegeneracy hypotheses) are glued together to produce new solutions. Although this sort of procedure is quite well-known, there have been some recent advances on which we wish to report here. We also discuss what has been established about the moduli spaces of all solutions to these problems, and report on some work in progress concerning global aspects of these moduli spaces. In the final section we present a new compactness result for the `unmarked moduli spaces' for problem III.
This is an introduction to the author's recent work on constrained systems. Firstly, a generalization of the Marsden-Weinstein reduction procedure in symplectic geometry is presented - this is a reformulation of ideas of Mikami-Weinstein and Xu. Secondly, it is shown how this procedure is quantized by Rieffel induction, a technique in operator algebra theory. The essential point is that a symplectic space with generalized moment map is quantized by a pre-(Hilbert) C^*-module. The connection with Dirac's constrained quantization method is explained. Three examples with a single constraint are discussed in some detail: the reduced space is either singular, or defined by a constraint with incomplete flow, or unproblematic but still interesting. In all cases, our quantization procedure may be carried out. Finally, we re-interpret and generalize Mackey's quantization on homogeneous spaces. This provides a double illustration of the connection between C^*-modules and the moment map.
This paper presents the construction of the Seiberg-Witten-Floer homology of three-manifolds with non-trivial rational homology, and some properties of the invariant of three-manifolds obtained by computing the Euler characteristic. This new version corrects a mistake and some misprints that appear in the published version (Internat. J. Math., Vol.7, N.5 (1996) 671-696). Various sections are rewritten in more detail.
In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an isolated fixed point. The motivation for this work comes from the program of classification of Hamiltonian group actions. The Audin-Ahara-Hattori-Karshon classification of Hamiltonian circle actions on compact symplectic 4-manifolds showed that all of such manifolds are Kaehler. Delzant's classification of $2n$-dimensional symplectic manifolds with Hamiltonian action of $n$-dimensional tori showed that all such manifolds are projective toric varieties, hence Kaehler. An example in this paper show that not all compact symplectic manifolds that admit Hamiltonian torus actions are Kaehler. Similar technique allows us to construct a compact symplectic manifold with a Hamiltonian circle action that admits no invariant complex structures, no invariant polarizations, etc.
Let $G_¶$ be a compact simple Poisson-Lie group equipped with a Poisson structure $¶$ and $(M, ø)$ be a symplectic manifold. Assume that $M$ carries a Poisson action of $G_¶$ and there is an equivariant moment map in the sense of Lu and Weinstein which acts to the dual Poisson-Lie group $G^*_¶$, $\m: M\rightarrow G^*_¶$. We prove that $M$ always possesses another symplectic form $\to$ so that the $G$-action preserves $\tildeø$ and there is a new moment map $μ= e^{-1} \circ \m: M\rightarrow \g^*$. Here $e$ is a universal (independent of $M$) invertible equivariant map $e: \g^*\rightarrow G^*_¶$. We suggest new short proves of the convexity theorem for the Poisson-Lie moment map, Poisson reduction theorem and the Ginzburg-Weinstein theorem on the isomorphism of $\g^*$ and $G^*_¶$ as Poisson spaces.
Let ${\cal D}^k$ be the space of $k$-th order linear differential operators on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural 1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal D}^k$. (To define this family, one considers arguments of differential operators as tensor-densities of degree $λ$.) In this paper we solve the problem of isomorphism between $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to different values of $λ$. The result is as follows: for $k=3$ $\Diff(\bf R)$-module structures on ${\cal D}^3$ are isomorphic to each other for every values of $λ\not=0,\;1,\;{1\over 2},\;{1\over 2}\pm \frac{\sqrt 21}{6}$, in this case there exists a unique (up to a constant) intertwining operator $T:{\cal D}^3\to{\cal D}^3$. In the higher order case $(k\geq 4)$ $\Diff(\bf R)$-module structures on ${\cal D}^k$ corresponding to two different values of the degree: $λ$ and $λ^{\prime}$, are isomorphic if and only if $λ+λ^{\prime}=1$.