Sphere Theorem Research Articles

Curvature, Sphere Theorems, and the Ricci flow

An innovative tool has been realized for the 3D/1D simulation of Ion Cyclotron Radio Frequency (ICRF), i.e. accounting for antennas in a realistic 3D geometry and with an accurate 1D plasma model. The approach to the problem is based on an integral-equation formulation for the self-consistent evaluation of the current distribution on the conductors. The environment has been subdivided in two coupled region: the plasma region and the vacuum region. The two problems are linked by means of a magnetic current (electric field) distribution on the aperture between the two regions. In the vacuum region all the calculations are executed in the spatial domain while in the plasma region an extraction in the spectral domain of some integrals is employed that permits to significantly reduce the integration support and to obtain a high numerical efficiency leading to the practical possibility of using a large number of sub-domain (rectangular or triangular) basis functions on each solid conductor of the system. The plasma enters the formalism of the plasma region via a surface impedance matrix; for this reason any plasma model can be used; at present the FELICE code has been adopted, that affords density and temperature profiles, and FLR effects. The source term directly models the TEM mode of the coax feeding the antenna and the current in the coax is determined self-consistently, giving the input impedance/admittance of the antenna itself. Calculation of field distributions (both magnetic and electric), useful for sheath considerations, is included. This tool has been implemented in a suite, called TOPICA, that is modular and applicable to ICRF antenna structures of arbitrary shape. This new simulation tool can assist during the detailed design phase and for this reason can be considered a Virtual Prototyping Laboratory (VPL). The TOPICA suite has been tested against assessed codes and against measurements and data of mock-ups and existing antennas. The VPL is being used in the design of various ICRF antennas and also for the performance prediction of the ALCATOR C-MOD D antenna.

Weyl connections and the local sphere theorem for quaternionic contact structures

We apply the theory of Weyl structures for parabolic geometries developed by Čap and Slovák (Math Scand 93(1):53–90, 2003) to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot–Carathéodory metric in the conformal class. The result of this computation has applications to the study of the conformal Fefferman space of a qc manifold, cf. (Geom Appl 28(4):376–394, 2010). In addition to this application, we are also able to easily compute a tensorial formula for the qc analog of the Weyl curvature tensor in conformal geometry and the Chern–Moser tensor in CR geometry. This tensor was first discovered via different methods by Ivanov and Vasillev (J Math Pures Appl 93:277–307, 2010), and we also get an independent proof of their Local Sphere Theorem. However, as a result of our derivation of this tensor, its fundamental properties—conformal covariance, and that its vanishing is a sharp obstruction to local flatness of the qc structure—follow as easy corollaries from the general parabolic theory.

A sphere theorem for pinching positive scalar curvature manifolds

Let ( M n , g ) , n ⩾ 3 , be a smooth closed Riemannian manifold with positive scalar curvature R g . There exists a positive constant C = C ( M , g ) , which is a geometric invariant, such that R g ⩽ n ( n − 1 ) C . In this paper we prove that R g = n ( n − 1 ) C if and only if ( M n , g ) is isometric to the Euclidean sphere S n ( C ) with constant sectional curvature C. Also, there exists a Riemannian metric g on M n such that the scalar curvature satisfies the pinched condition, n 2 ( n − 2 ) n − 1 C < R g ⩽ n ( n − 1 ) C , if and only if M n is diffeomorphic to the standard sphere S n .

An Integral Formula for Lipschitz-Killing Curvature and the Critical Points of Height Functions

We have established (see Shiohama and Xu in J. Geom. Anal. 7:377–386, 1997; Lemma) an integral formula on the absolute Lipschitz-Killing curvature and critical points of height functions of an isometrically immersed compact Riemannian n-manifold into Rn+q. Making use of this formula, we prove a topological sphere theorem and a differentiable sphere theorem for hypersurfaces with bounded Ln/2 Ricci curvature norm in Rn+1. We show that the theorems of Gauss-Bonnet-Chern, Chern-Lashof and the Willmore inequality are all its consequences.

Riemannian Manifolds of Positive Curvature

The study of positive sectional curvature is one of the oldest pursuits in Riemannian geometry, but despite the considerable efforts of many researchers, basic questions remain unanswered. In this lecture we will briefly summarize the state of knowledge in this area and outline the techniques which have had success. These techniques include geodesic and comparison methods, minimal surface methods, and Ricci flow. We will then describe our recent work (see [18], [21], [22]) which uses the Ricci flow to resolve the differentiable sphere theorem; that is, the complete classification of manifolds whose sectional curvatures are 1/4-pinched.

Positively Curved Combinatorial 3-Manifolds

We present two theorems in the "discrete differential geometry" of positively curved spaces. The first is a combinatorial analog of the Bonnet-Myers theorem: $\bullet$ A combinatorial 3-manifold whose edges have degree at most five has edge-diameter at most five. When all edges have unit length, this degree bound is equivalent to an angle-deficit along each edge. It is for this reason we call such spaces positively curved. Our second main result is analogous to the sphere theorems of Toponogov and Cheng: $\bullet$ A positively curved 3-manifold, as above, in which vertices $v$ and $w$ have edge-distance five is a sphere whose triangulation is completely determined by the structure of $Lk(v)$ or $Lk(w)$. In fact, we provide a procedure for constructing a maximum diameter sphere from a suitable $Lk(v)$ or $Lk(w)$. The compactness of these spaces (without an explicit diameter bound) was first proved via analytic arguments in a 1973 paper by David Stone. Our proof is completely combinatorial, provides sharp bounds, and follows closely the proof strategy for the classical results.

Extension theorems for spheres in the finite field setting

In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields. In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, previous results by the authors in [Iosevich and Koh, Illinois J. of Mathematics: 2008]. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums.

A topological sphere theorem for arbitrary-dimensional manifolds

We study manifolds with bounded volume, sectional curvature, and injectivity radius. We obtain a topological sphere theorem.

Conformal quaternionic contact curvature and the local sphere theorem

A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern–Moser tensor in CR geometry. It is shown that a quaternionic contact manifold is locally quaternionic contact conformal to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-Sasakian structure on the sphere iff the quaternionic contact conformal curvature vanishes.

Diameter pinching in almost positive Ricci curvature

In this paper we prove a diameter sphere theorem and its corresponding λ1 sphere theorem under Lp control of the curvature. They are generalizations of some results due to S. Ilias [8].

Liouville theorems for self-similar solutions of heat flows

Let N be a compact Riemannian manifold. A quasi-harmonic sphere is a harmonic map from ({\bf R}^m, e^{-|x|^2/2(m-2)}ds_0^2) to N ( m\geq 3 ) with finite energy ([LnW]). Here ds_0^2 is the Euclidean metric in {\bf R}^m . It arises from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target N . We also derive gradient estimates and Liouville theorems for positive quasi-harmonic functions.

A rigidity phenomenon on riemannian manifolds with reverse excess pinching

The authors introduce the Hausdorff convergence to discuss the differentiable sphere theorem with excess pinching. Finally, a type of rigidity phenomenon on Riemannian manifolds is derived.

Topological and differentiable sphere theorems for complete submanifolds

Topological and differentiable sphere theorems for complete submanifolds

VANISHING AND TOPOLOGICAL SPHERE THEOREMS FOR SUBMANIFOLDS IN A HYPERBOLIC SPACE

We extend the vanishing and sphere theorems due to Lawson, Simons, Xin, Shiohama and Xu. By using the techniques of calculus of variations in the geometric measure theory, we prove the vanishing theorem for homology groups of submanifolds in the hyperbolic space Hn(c) with negative constant curvature c. Moreover, we obtain a topological sphere theorem for certain complete submanifolds in Hn(c).

Partielle Differentialgleichungen

The workshop differential equations , organised by Tom Ilmanen (ETH Zürich), Reiner Schätzle (Universität Tübingen) and Neil Trudinger (Australian National University Canberra) was held July 22–28, 2007. This meeting was well attended by 50 participants, including 6 females, with broad geographic reprensentation. The program consisted of 19 talks and 3 shorter contributions and left sufficient time for discussions. One focus was the combination of nonlinear partial differential equations and geometric problems. There were talks on monotonicity formulas in presence of degenerate singularities and on estimates of the singular set for branched minimal immersions. One striking new result was the proof of the diffeomorphic sphere theorem by the use of Ricci flow. Many leading experts in the regularity of elliptic and variational problems and in fully nonlinear differential equations occuring in optimal transport and in Hessian equations attended the workshop. Several main contributions were given to this subject. Here, we mention the proof of interior estimates for quadratic Hessian equations in three dimensions, which has been an open problem for several years of similar relevance as the gradient estimates for the minimal surface equation. New results were also presented for the Willmore functional. It was achieved to write the Euler-Lagrange equation in divergence form, and the existence of minimizers in fixed conformal classes with certain energy bounds were obtained. The organisers and the participants are grateful to the Oberwolfach Institute for presenting the opportunity and the resources to arrange this interesting meeting.

A sphere theorem for a class of Reinhardt domains with constant Levi curvature

We prove that a bounded Reinhardt domain of ℂn+1 with C2 boundary of constant Levi curvature with rotational symmetry in two complementary sets of variables is a ball.

Radius Sphere Theorems for Compact Manifolds with Radial Curvature Bounded Below

We show a radius sphere theorem for a compact Riemannian manifold whose radial curvature at the base point is bounded from below by that of a 2-sphere of revolution. The diameter sphere theorem is expanded to a wide class of metrics.

A rigidity phenomenon on Riemannian manifolds with reverse volume pinching

In the article stated here, we introduce the Hausdorff convergence to discuss the differentiable sphere theorem with some curvature conditions and volume pinching. Finally, a type of rigidity phenomenon on Riemannian manifolds will be derived.

Topology of Complete Manifolds with Radial Curvature Bounded from Below

We investigate the topology of a complete Riemannian manifold whose radial curvature at the base point is bounded from below by that of a von Mangoldt surface of revolution. Sphere theorem is generalized to a wide class of metrics, and it is proven that such a manifold of a noncompact type has finitely many ends.

A differentiable sphere theorem on manifolds with reverse volume pinching

Based on the celebrated 1/4-pinching sphere theorem, we prove a differentiable sphere theorem on Riemannian manifolds with reverse volume pinching.