Let N be a closed connected spin manifold admitting one metric of positive scalar curvature. In this paper we use the higher eta-invariant associated to the Dirac operator on N in order to distinguish metrics of positive scalar curvature on N . In particular, we give sufficient conditions, involving π1(N) and dim N , for N to admit an infinite number of metrics of positive scalar curvature that...

We consider the Dolbeault operator √ 2(∂ + ∂ *) of K 1 2 – the square root of the canon-ical line bundle which determines the spin structure of a compact Hermitian spin surface (M, g, J). We prove that all cohomology groups H 1 2)) vanish if the scalar curvature of g is non-negative and non-identically zero. Moreover, we estimate the first eigenvalue of the Dolbeault operator when the conformal...

For a submanifold Σ ⊂ R Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds Σ in the same way. More generally, on any metric measure we are able to approximate a 1-parameter family of coarse Ricci functions that include ...

Abstract. We apply the Piecewise Constant Level Set Method (PCLSM) to a multiphase motion problem, especially the pure mean curvature motion. We use one level set function to represent multiple regions, and by associating an energy functional which consists of surface tension (proportional to length), we formulate a variational approach for the mean curvature motion problem. Some operator-split...

In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with 2-nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension ≥ 3) whose curvature operator is bounded and satisfies the pinching condition ...

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds (M, g) for the following general heat equation ut = ∆V u+ au log u+ bu where a is a constant and b is a differentiable function defined onM×[0,∞). We suppose that the Bakry-Émery curvature and the N -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimat...

Background: Scoliosis is the most common type of spinal deformity. A universal and standard method for evaluating scoliosis is Cobb angle measurement, but several studies have shown that there is intra- and inter- observer variation in measuring cobb angle manually.Objective: Develop a computer- assisted system to decrease operator-dependent errors in Cobb angle measurement.Methods: The spinal ...

We study the quantum sphere Cq [S] as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2-dimensional cotangent bundle as a direct sum Ω ⊕ Ω in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators. We show that the q-monopole as spin connection induces a natural Levi-Civita type connection and find its Ricci ...

For an operator T in the class Bn(Ω), introduced by Cowen and Douglas, the simultaneous unitary equivalence class of the curvature and the covariant derivatives up to a certain order of the corresponding bundle ET determine the unitary equivalence class of the operator T . In a subsequent paper, the authors ask if the simultaneous unitary equivalence class of the curvature and these covariant d...

We obtain a vanishing theorem for the kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the kernel of a spinc Dirac operator twisted by a line bundle with curvature of a mixed sign. In t...