Localized gluing of Riemannian metrics in interpolating their scalar curvature

We show that two smooth nearby Riemannian metrics can be glued interpolating their scalar curvature. The resulting smooth metric is the same as the starting ones outside the gluing region and has scalar curvature interpolating between the original ones. One can then glue metrics while maintaining inequalities satisfied by the scalar curvature. We also glue asymptotically Euclidean metrics to Schwarzschild ones and the same for asymptotically Delaunay metrics, keeping bounds on the scalar curvature, if any. This extends the Corvino gluing near infinity to non-constant scalar curvature metrics.     

Remark about scalar curvature and Riemannian submersions

We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space and fibers. We give an application concerning scalar curvatures of smooth limit spaces arising in bounded curvature collapses.

     

Estimates and nonexistence of solutions of the scalar curvature equation on noncompact manifolds

This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.     

An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Ricci \ge 0

By establishing an optimal comparison result on the heat kernel of the conformal Laplacian on open manifolds with nonnegative Ricci curvature, (a) we show that many manifolds with positive scalar curvature do not possess conformal metrics with scalar curvature bounded below by a positive constant; (b) we identify a class of functions with the following property: If the manifold has a scalar curvature in this class, then there exists a complete conformal metric whose scalar curvature is any given function in this class. This class is optimal in some sense; (c) we have identified all manifolds with nonnegative Ricci curvature, which are “uniformly” conformal to manifolds with zero scalar curvature. Even in the Euclidean case, we obtain a necessary and sufficient condition under which the main existence results in [Ni1] and [KN] on prescribing nonnegative scalar curvature will hold. This condition had been sought in several papers in the last two decades. Received: 11 November 1998 / Revised: 7 April 1999     

On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.     

An Equivalence of Scalar Curvatures on Hermitian Manifolds

For a Kähler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kähler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu–Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kähler. However, we prove that a certain class of non-compact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.     

Non‐Gaussian invariant measures for the Majda model of decaying turbulent transport

The problem of turbulent transport of a scalar field by a random velocity field is considered. The scalar field amplitude exhibits rare but very large fluctuations whose typical signature are fatter than Gaussian tails for the probability distribution of the scalar. The existence of such large fluctuations is related to clustering phenomena of the Lagrangian paths within the flow. This suggests an approach to turn the large deviation problem for the scalar field into a small deviation, or small ball, problem for some appropriately defined process measuring the spreading with time of the Lagrangian paths. Here, such a methodology is applied to a model proposed by Majda consisting of a white‐in‐time linear shear flow and some generalizations of it where the velocity field has finite, or even infinite, correlation time. The non‐Gaussian invariant measure for the (reduced) scalar field is derived and, in particular, it is shown that the one‐point distribution of the scalar has stretched exponential tails, with a stretching exponent depending of the parameters in the model. Different universality classes for the scalar behavior are identified which, all other parameters being kept fixed, display a one‐to‐one correspondence with a exponent measuring time persistence effects in the velocity field. © 2001 John Wiley & Sons, Inc.     

Blowing up and desingularizing constant scalar curvature Kähler manifolds

This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow-ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature Kähler metrics.     

Riemannian metrics of positive scalar curvature on compact manifolds with boundary

We prove that any metric of positive scalar curvature on a manifold X extends to the trace of any surgery in codim > 2 on X to a metric of positive scalar curvature which is product near the boundary. This provides a direct way to construct metrics of positive scalar curvature on compact manifolds with boundary. We also show that the set of concordance classes of all metrics with positive scalar curvature on S ⁿ is a group.     

The scalar curvature flow on <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>—perturbation theorem revisited

For the problem of finding a geometry on S ⁿ , for n≥3, with a prescribed scalar curvature, there is a well-known result which is called the perturbation theorem; it is due to Chang and Yang (Duke Math. J. 64, 27–69, 1991). Their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm. It is important to know how large that difference in sup norm can possibly be. Here we consider prescribing scalar curvature problem using the scalar curvature flow.     

局部对称流形上的数量曲率

本文讨论了无共轭点测地线上的Ｊａｃｏｂｉ声，证明了具非负数量曲率的局部对称的无共轭点流形及具非负数量曲率的具极点的局部对称的流形之数量曲率只能是零。部分解决了Ｅ．Ｈｏｐｆ猜想。     

On complete Finslerian Yamabe solitons

Here it is shown that any Finslerian compact Yamabe soliton with bounded above scalar curvature is of constant scalar curvature. Furthermore, this extension of Yamabe solitons is developed for inequalities and among the others, it is proved that a forward complete non-compact shrinking Yamabe soliton has finite fundamental group and its first cohomology group vanishes, providing the scalar curvature is strictly bounded above.     

Properties of Berwald scalar curvature

We prove that a Finsler manifold with vanishing Berwald scalar curvature has zero E-curvature. As a consequence, Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds. For (α,β)-metrics on manifold of dimension greater than 2, if the mean Landsberg curvature and the Berwald scalar curvature both vanish, then the Berwald curvature also vanishes.     

The modified Yamabe problem and geometry of modified scalar curvatures

Study of modified scalar curvatures and the modified Yamabe problem is given. An existence theorem of type of Trudinger and Aubin is obtained. Interpreting Isotropic curvature as a certain modified scalar curvature we can extend the four-dimensional sphere theorem of [23]. Another modified scalar curvature is used in solving open problems concerning the conjecture of self-dual Einstein 4-manifold and the Goldberg conjecture, both being under negative Ricci condition.     

Classification of ordinary differential equations by conditional linearizability and symmetry

Lie’s invariant criteria for determining whether a second order scalar equation is linearizable by point transformation have been extended to third and fourth order scalar ordinary differential equations (ODEs). By differentiating the linearizable by point transformation scalar second order ODE (respectively third order ODE) and then requiring that the original equation holds, what is called conditional linearizability by point transformation of third and fourth order scalar ODEs, is discussed. The result is that the new higher order nonlinear ODE has only two arbitrary constants available in its solution. One can use the same procedure for the third and fourth order extensions mentioned above to get conditional linearizability by point or other types of transformation of higher order scalar equations. Again, the number of arbitrary constants available will be the order of the original ODE. A classification of ODEs according to conditional linearizability by transformation and classifiability by symmetry are proposed in this paper.     

Quantum mechanical model in gravity theory

We consider a model of a real massive scalar field defined as homogeneous on a d-dimensional sphere such that the sphere radius, time scale, and scalar field are related by the equations of the general theory of relativity. We quantize this system with three degrees of freedom, define the observables, and find dynamical mean values of observables in the regime where the scalar field mass is much less than the Planck mass.     

On NP-Completeness for Linear Machines

We consider linear and scalar versions of the Blum–Shub–Smale model of computation over the reals. The permitted computing operations of linear machines are addition and multiplication by constants. The scalar machines can only multiply by constants. The size of an input is its dimension, and the cost of any instruction is one. For each of these structures we consider DNP and NP, the corresponding complexity classes with respect to digital nondeterminism and standard real nondeterminism, respectively. We give DNP- and NP-complete problems for linear and real scalar machines. On the other hand, we show that the NP-class restricted to scalar machines over the integers with equality-tests does not own a complete problem.     

Flexible isosurfaces: Simplifying and displaying scalar topology using the contour tree

The contour tree is an abstraction of a scalar field that encodes the nesting relationships of isosurfaces. We show how to use the contour tree to represent individual contours of a scalar field, how to simplify both the contour tree and the topology of the scalar field, how to compute and store geometric properties for all possible contours in the contour tree, and how to use the simplified contour tree as an interface for exploratory visualization.     

Remarks on scalar curvature of Yamabe solitons

In this article, we consider the scalar curvature of Yamabe solitons. In particular, we show that, with natural conditions and non-positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that a complete non-compact Yamabe soliton with the quadratic decay at infinity of its Ricci curvature has non-negative scalar curvature. A new proof of Kazdan?CWarner condition is also presented.     

On order of elements in hypergroups and p-Sylow subhypergroups

In this paper we define a left order, right order and order of an element in a hypergroup, which has a left scalar identity, right scalar identity and scalar identity, respectively. Using left and right orders, we define and investigate some classes of subhypergroups, especially p-Sylow subhypergroups of a hypergroup. Moreover, some examples have been proposed to show that the Sylow theorems do not hold for polygroups.