Combinatorial handles and manifolds with boundary

In this work we complete the study ofcombinatorial handles in (n+1)-coloured graphs with boundary, introduced in [G₁], [L] and [GV] for graphs with empty boundary and in [BGV] for 3-coloured graphs with boundary. In particular, we study the cancelling of a combinatorial handle from an (n+1)-coloured graph and its effects on the associated complex.Work performed under the auspices of the G.N.S.A.G.A. — C.N.R., and within the Project Geometria reale e complessa, supported by M.U.R.S.T. of Italy.     

Isoperimetric comparison theorems for manifolds with density

We give several isoperimetric comparison theorems for manifolds with density, including a generalization of a comparison theorem from Bray and Morgan. We find for example that in the Euclidean plane with radial density exp(r ^α ) for α ≥ 2, discs about the origin minimize perimeter for given area, by comparison with Riemannian surfaces of revolution.     

Isoperimetric profile comparisons and Yamabe constants

We estimate from below the isoperimetric profile of S² ×\mathbb R²{S^2 \times {\mathbb R}^2} and use this information to obtain lower bounds for the Yamabe constant of S² ×\mathbb R²{S^2 \times {\mathbb R}^2} . This provides a lower bound for the Yamabe invariants of products S ² ×  M ² for any closed Riemann surface M. Explicitly we show that Y (S ² ×  M ²) >  (2/3)Y(S ⁴).     

Generalized handles in graphs and connected sums of manifolds

Generalized handles in (n+1)-coloured graphs are defined. They are the n-dimensional analogue of a concept listed in [19] and generalizes the definition of combinatorial handle first introduced in [8] and successively investigated in[2],[12],[19],[26]. Then the operations of cancelling and cutting a generalized handle are studied. As a consequence some decomposition theorems about manifolds are proved. In particular, a graph-theoretical condition is obtained to recognize S^n–1 -bundles over S¹ as factors of a decomposition of a manifold in connected sum.Work performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council of Italy) and financially supported by the M.P.I. of Italy within the Project Geometria delle Variet differenziabili     

Seshadri constants and Fano manifolds

In memory of Meeyoung Kim In this paper, we give a lower bound of Seshadri constants on smooth Fano varieties. More precisely, we show that on a smooth Fano manifold of dimension n whose anticanonical system is base point free, Seshadri constants of ample divisors are bounded from below by one over n–2. As a corollary we recover the earlier result on Fano threefolds. Mathematics Subject Classification (2000):14J45, 14N30.The author was supported in part by KOSEF Grant R14-2002-007-01001-0(2002).     

Projective manifolds with small pluridegrees

Let be a very ample line bundle on a connected complex projective manifold of dimension . Except for a short list of degenerate pairs , and there exists a morphism expressing as the blowup of a projective manifold at a finite set , with nef and big for the ample line bundle . The projective geometry of is largely controlled by the pluridegrees for , of . For example, , where is the genus of a curve section of , and is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of . In this article, a detailed analysis is made of the pluridegrees of . The restrictions found are used to give a new lower bound for the dimension of the space of sections of . The inequalities for the pluridegrees, that are presented in this article, will be used in a sequel to study the sheet number of the morphism associated to .

     

Closed manifolds with small excess

In this article, we study closed Riemannian manifolds with small excess. We show that a closed connected Riemannian manifold with Ricci curvature and injectivity radius bounded from below is homeomorphic to a sphere if it has sufficiently small excess. We also show that a closed connected Riemannian manifold with weakly bounded geometry is a homotopy sphere if its excess is small enough.     

Pinching constants for hyperbolic manifolds

Summary We show in this paper that for everyn4 there exists a closedn-dimensional manifoldV which carries a Riemannian metric with negative sectional curvatureK but which admits no metric with constant curvatureK–1. We also estimate the (pinching) constantsH for which our manifoldsV admit metrics with –1K–H.     

Isoperimetric constants and estimates of heat kernels of pre Sierpinski carpets

The author calculated isoperimetric constants of then-dimensional pre Sierpinski carpetY _n . As an application, he obtained the following estimate of the Neumann heat kernelp _n (t, x, y) onY _n ; where

Asymptotics of best Sobolev constants on thin manifolds

We study the asymptotic behaviour of best Sobolev constants on a compact manifold with boundary that we contract in k directions or to a point. We find in the limit best Sobolev constants for weighted Sobolev spaces of the limit manifold.     

Three-dimensional manifolds of nonpositive curvature with small injectivity radius

It is proved that if at every point of a closed, three-dimensional, Riemannian manifold with bounded sectional curvature the injectivity radius does not exceed a specific absolute constant, then the manifold is a special graph and its metric splits locally.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 10–19, 1991.     

Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension

For compact Riemannian manifolds with convex boundary, B. White proved the following alternative: either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B. White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four–dimensional ball B ⁴ with the following properties: (i) B ⁴ has strictly convex boundary. (ii) There exists a complete nonconstant geodesic ${c : \mathbb{R} \to B^4}$ . (iii) There does not exist a closed geodesic in B ⁴.     

Sharp constants in weighted trace inequalities on Riemannian manifolds

In this paper, we establish some sharp weighted trace inequalities ${W^{1,2}(\rho^{1-2 \sigma}, M) \hookrightarrow L^{\frac{2n}{n-2 \sigma}}(\partial M)}$ on n + 1 dimensional compact smooth manifolds with smooth boundaries, where ρ is a defining function of M and ${\sigma \in (0,1)}$ . This is stimulated by some recent work on fractional (conformal) Laplacians and related problems in conformal geometry, and also motivated by a conjecture of Aubin.     

Cohomogeneity one manifolds with a small family of invariant metrics

In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible symmetric space. As a corollary such manifolds admit an invariant metric with non-negative sectional curvature.     

Non-periodic Riemann examples with handles

We show the existence of 1-parameter families of non-periodic, complete, embedded minimal surfaces in Euclidean space with infinitely many parallel planar ends. In particular we are able to produce finite genus examples and quasi-periodic examples of infinite genus.     

Inertial and slow manifolds for delay equations with small delays

Yu.A. Ryabov and R.D. Driver proved that delay equations with small delays have Lipschitz inertial manifolds. We prove that these manifolds are smooth. In addition, we show that expansion in the small delay can be used to obtain the dynamical system on the inertial manifold. This justifies “post-Newtonian” approximation for delay equations.     

Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature

Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 2, let p(1, n) real, and let H₁^p (M) be the standard Sobolev space of order p. By the Sobolev embedding theorem, H₁^p(M) ⊂ L^p* (M) where p^* = np/(n - p). Classically, this leads to some Sobolev inequality (I_p¹), and then to some Sobolev inequality (I_p^p) where each term in (I_p¹) is elevated to the power p. Long standing questions were to know if the optimal versions with respect to the first constant of (I_p¹) and (I_p^p) do hold. Such questions received an affirmative answer by Hebey-Vaugon for p = 2, and on what concerns (I_p¹), by Aubin for two-dimensional manifolds and for manifolds of constant sectional curvature. Recently, Druet proved that for p > 2, and p² < n, the optimal version of (I_p^p) is false if the scalar curvature of g is positive somewhere, while for p > 1, the optimal version of (I_p^p) does hold on flat torii and compact hyperbolic spaces. We prove here that the optimal version of (I_p^p), p > 1, does hold for compact manifolds of nonpositive sectional curvature in any dimension where the Cartan-Hadamard conjecture is true. In particular, since the Cartan-Hadmard conjecture is true in dimensions 2, 3, and 4, the optimal version of (I_p^p) does hold on any compact manifold of nonpositive sectional curvature of dimension 2, 3, or 4.