Showing papers in "Journal of Differential Geometry in 1982"

Supersymmetry and Morse theory

Abstract: It is shown that the Morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics Hamiltonian. Some of the implications of modern ideas in mathematics for supersymmetric theories are discussed.

The topology of four-dimensional manifolds

Abstract: 0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's. Of the mysteries still remaining after that period of great success the most compelling seemed to lie in dimensions three and four. Although experience suggested that manifold theory at these dimensions has a distinct character, the dream remained since my graduate school days that some key principle from the high dimensional theory would extend, at least to dimension four, and bring with it the beautiful adherence of topology to algebra familiar in dimensions greater than or equal to five. There is such a principle. It is a homotopy theoretic criterion for imbedding (relatively) a topological 2-handle in a smooth four-dimensional manifold with boundary. The main impact, as outlined in §1, is to the classification of 1-connected 4-manifolds and topological end recognition. However, certain applications to nonsimply connected problems such as knot concordance are also obtained. The discovery of this principle was made in three stages. From 1973 to 1975 Andrew Casson developed his theory of "flexible handles". These are certain pairs having the proper homotopy type of the common place open 2-handle H = (D X D, dD X D) but "flexible" in the sense that finding imbeddings is rather easy; in fact imbedding is implied by a homotopy theoretic criterion. It was clear to Casson that: (1) no known invariant—link theoretic

Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds

Abstract: where dEλ is the projection valued measure associated with /^Δ\". A natural problem is to study the behavior of the explicit kernel kf(X)(xx, x2) representing /(/^Δ), in terms of the behavior of various geometric quantities on M. As a particularly important example we have the heat kernel E(xl9 x2, t) — ke-\\2t. By use of the local parametrix and the standard elliptic estimates, one can show that for / > 0, E(xλ9 JC2, 0 is a positive (symmetric) C function of JC1? x2, t which for fixed t and (say) %2> ι s * the domain of all positive powers of Δ as a function of xλ; see e.g. [9]. In works of Garding [19] and Donnelly [16], upper estimates for E(xu x2, t) (and its derivatives) were given under the assumption that M has bounded geometry. They showed that as x2 -> oo, the behavior of E{xλ, x2, t) is roughly similar to that of the e-p 2(xx,x2)/4 Euclidean heat kernel, — (p(xx, x2) denotes distance). Recall that (4ττ/) M is said to have bounded geometry if the injectivity radius i(x) of the

A regularity theory for harmonic maps

Abstract: 0. Introduction In this paper we develop a regularity theory for energy minimizing harmonic maps into Riemannian manifolds. Let u: M -> N be a map between Riemannian manifolds of dimension n and k. It was shown by C. B. Morrey [17] in 1948 that if n — 2, then an energy minimizing harmonic map is Holder continuous (and smooth if M and N are smooth). Since that time results have been found under special assumptions on N. Eells and Sampson [5] proved in 1963 that if N is compact and has nonpositive curvature, then every homotopy class of maps from a closed manifold M into N has a smooth harmonic representative. In the case where the image of the map is contained in a convex ball of N9 there is a complete existence and regularity theory due to Hildebrandt and Widman [15] as well as Hildebrandt, Kaul and Widman [13]. Recently Giaquinta and Giusti obtained results for the case in which the image lies in a coordinate chart [9], [10]. In this paper we show that a bounded, energy minimizing map u: M -» N is regular (in the interior) except for a closed set S of Hausdorff dimension at most n — 3. We also show S is discrete for n = 3. Moreover, we derive techniques (see Theorem IV) for lowering the dimension of S under the condition that certain smooth harmonic maps of spheres into N are trivial. This can be checked in some interesting cases, for example if N has nonpositive curvature or if the image of the map lies in a convex ball of N, we show S = 0 and any minimizing harmonic map into such a manifold is smooth. Using our methods, it is possible to reduce the dimension of S if N is a sphere or Lie group by studying harmonic spheres in N. Our methods work for functional which are the energy plus lower order terms, and thus have direct bearing on the question of the existence of global Coulomb gauges in nonabelian gauge theories. We point out that there is a strong historical precedent for partial regularity results in problems involving elliptic systems (see Almgren [1], De Giorgi [3],

A lower bound for the first eigenvalue of a negatively curved manifold

Abstract: There has been much work in recent years on the relation of the low eigenvalues of a compact Riemannian manifold to the geometry of the manifold. For Riemann surfaces with positive genus, it was observed by P. Buser [1] that one can find a compact hyperbolic surface of fixed genus (hence fixed area) with arbitrarily small first eigenvalue (see [10] for more information on this problem). For hyperbolic manifolds of dimension larger than two, Mostow's theorem implies that the topology uniquely determines the geometry, so the above phenomenon for λ! is likely to be a two-dimensional phenomenon. In this note we show that this is the case. Precisely, let M\" be a compact Riemannian manifold with sectional curvature bounded between two negative constants. We show here that if n > 3, then λx(M) has a lower bound depending only on the volume of M. Actually, for n > 3, Gromov [7] has shown that an upper bound on volume implies an upper bound on diameter (for negatively curved M). Using this result, a bound such as ours would follow from a general result of S. T. Yau [11]. For n — 3, the diameter is not bounded in terms of volume (see [2, 3.13]) so our result seems to be of most interest in this case. Buser [2] has observed that our dependence on the inverse square of the volume is best possible. The case n = 3 of our theorem was announced in the Hawaii Symposium in 1979. In this note we give a simplified version, valid for all n > 2, of our original proof. We wish to thank P. Buser for pointing out reference [9] which is used in the proof of Lemma 1.