Complete Riemannian metrics with holonomy group G 2 on deformations of cones over S 3 × S 3

Complete Riemannian metrics with holonomy group G ₂ on manifolds obtained by deformation of cones over S ³ × S ³ are constructed.     

Nielsen theory on 3-manifolds covered by S 2 × ℝ

Let f: M → M be a self-map of a closed manifold M of dimension dim M ≥ 3. The Nielsen number N(f) of f is equal to the minimal number of fixed points of f among all self-maps f in the homotopy class of f. In this paper, we determine N(f) for all self-maps f when M is a closed 3-manifold with S²×R geometry. The calculation of N(f) relies on the induced homomorphisms of f on the fundamental group and on the second homotopy group of M.     

Nonnegatively Curved Metrics on S 2 × ℝ2

We classify the complete metrics of nonnegative sectional curvature on M ² × ², where M ² is any compact 2-manifold.     

On asymptotics of complete Ricci-flat Kähler metrics on open manifolds

Tian and Yau constructed in J. Am. Math. Soc., 3(3):579–609, 1990, a complete Ricci-flat Kähler metric on the complement of an ample and smooth anticanonical divisor. For the explicitly constructed referential metric ω of Tian and Yau (J. Am. Math. Soc., 3(3):579–609, 1990) we prove a property that ${\|\partial\overline\partial u\|_\omega}$ has the same decay rate as Δ _ω u provided u satisfies some decay conditions on higher Laplacians. As an application we describe the behaviour of this metric towards the boundary divisor and prove the best possible decay rate of the difference to ω.     

On the Yamabe constants of S2×R3 and S3×R2

We compare the isoperimetric profiles of $S^2\times {\mathbb{R}}^3$ and of $S^3\times {\mathbb{R}}^2$ with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of $S^2\times {\mathbb{R}}^3$ and $S^3\times {\mathbb{R}}^2$. Explicitly we show that $Y(S^3\times {\mathbb{R}}^2,[g_0^3+d{x}^2])>(3/4)Y(S^5)$ and $Y(S^2\times {\mathbb{R}}^3,[g_0^2+d{x}^2])>0.63Y(S^5)$. We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in Petean and Ruiz (2011) [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions.     

On the existence of complete Kähler metrics of negative Riemannian curvature bounded away from zero on ellipsoidal domains inC n

Summary The purpose of this paper is to prove that every ellipsoidal domain in Cⁿ admits a complete Kähler metric whose Riemannian sectional curvature is bounded from above by a negative constant (Theorem 1). We construct a Kähler metric, in a natural way, as potential of a suitable function defining the boundary (§2). Directly we compute the curvature tensor and we find upper and lower bounds for the holomorphic sectional curvature (§ 4, § 5). In order to prove the boundness of Riemannian sectional curvature we use finally a classical pinching argument (§ 6). We also obtain that for certain ellipsoidal domains the curvature tensor is very strongly negative in the sense of [15] (§ 3). Finally we prove that the metric constructed on ellipsoidal domains in Cⁿ is the Bergman metric if and only if the domain is biholomorphic to the ball (Theorem 2). In [8], [9] R. E. Greene and S. G. Krantz gave large families of examples of complete Kähler manifolds with Riemannian sectional curvature bounded from above by a negative constant; they are sufficiently small deformations of the ball in Cⁿ, with the Bergman metric. Before the only known example of complete simply-connected Kähler manifold with Riemannian sectional curvature upper bounded by a negative constant, not biholomorphic to the ball, was the surface constructed by G. D. Mostow and Y. T. Siu in [14], to the best of the author's knowledge, is not known at present if this example is biholomorphic to a domain in Cⁿ.     

On a class of weak Landsberg metrics

In this paper, we discuss a class of Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We characterize weak Landsberg metrics in this class and show that there exist weak Landsberg metrics which are not Landsberg metrics in dimension greater than two.     

Free involutions on S 2 × S 3

In this paper, we classify smooth 5-manifolds with fundamental group isomorphic to ${\mathbb{Z}/2}$ and universal cover diffeomorphic to S ² × S ³. This gives a classification of smooth free involutions on S ² × S ³ up to conjugation.     

On G 2 Continuous Spline Interpolation of Curves in ℝ d

In this paper the problem of G ² continuous interpolation of curves in ^d by polynomial splines of degree n is studied. The interpolation of the data points and two tangent directions at the boundary is considered. The case n = r + 2 = d, where r is the number of interior points interpolated by each segment of the spline curve, is studied in detail. It is shown that the problem is uniquely solvable asymptotically, e., when the data points are sampled regularly and sufficiently dense, and lie on a regular, convex parametric curve in ^d . In this case the optimal approximation order is also determined.     

Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of minimal herissons in ³ is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC ³ are described by a single map from the étalé space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ₁ to C³, which islinear on stalks. There is an analogous construction for null curves inC ⁴. This gives a similar class of minimal surfaces in ⁴.     

On uniqueness of Frankl-Morawetz problem in ℝ2

For the equation $$L[u]: = K(y)u_{xx} + u_{yy} + r(x,y)u = f(x,y)$$ (K (y)?0 whenevery?0) inG, bounded by a piecewise smooth curveΓ ₀ fory>0 which intersects the liney=0 at the pointsA(?1, 0) andB(1, 0) and fory<0 by a smooth curveΓ ₁ throughA which meets the characteristic of (1) throughB at the pointP, the uniqueness of the Frankl-Morawetz problem is proved without assuming thatΓ ₁ is monotone.     

On the rank of 3×3×3-tensors

Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U???V???W is the minimum dimension of a subspace of U???V???W containing τ and spanned by fundamental tensors, i.e. tensors of the form u???v???w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U???V???W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U???V???W when the dimensions of U, V and W are higher.     

On the weak L p -Hodge decomposition and Beurling�CAhlfors transforms on complete Riemannian manifolds

Based on a new martingale representation formula, we prove some quantitative upper bound estimates of the L ^p -norm of some singular integral operators on complete Riemannian manifolds. This leads us to establish the Weak L ^p -Hodge decomposition theorem and to prove the L ^p -boundedness of the Beurling?CAhlfors transforms on complete non-compact Riemannian manifolds with non-negative Weitzenb?ck curvature operator.     

Isometric embeddings of locally Euclidean metrics in ℝ3 as conical surfaces

It is proved that if a domain with a locally Euclidean metric can be isometrically immersed in the Euclidean plane ?² with the standard metric, then it can be isometrically embedded in ?³ as a conical surface whose projection on a sphere centered at the vertex of the cone is a self-avoiding planar graph with sufficiently smooth edges of specially selected lengths.