Conjugate Radius and Sphere Theorem

1999 ◽  
Vol 42 (2) ◽  
pp. 214-220 ◽  
Author(s):  
Seong-Hun Paeng ◽  
Jong-Gug Yun
Keyword(s):  
Lower Bound ◽  
Injectivity Radius ◽  
Sphere Theorem ◽  
Standard Sphere

AbstractBessa [Be] proved that for given n and i0, there exists an depending on n, i0 such that if M admits a metric g satisfying , then M is diffeomorphic to the standard sphere. In this note, we improve this result by replacing a lower bound on the injectivity radius with a lower bound of the conjugate radius.

scholarly journals Geometry of shrinking Ricci solitons

2015 ◽  
Vol 151 (12) ◽  
pp. 2273-2300 ◽  
Author(s):  
Ovidiu Munteanu ◽  
Jiaping Wang
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Scalar Curvature ◽  
Distance Function ◽  
Upper Bound ◽  
Arbitrary Dimension ◽  
Injectivity Radius ◽  
Curvature Operator ◽  
Ricci Solitons ◽  
Gradient Ricci Solitons

The main purpose of this paper is to investigate the curvature behavior of four-dimensional shrinking gradient Ricci solitons. For such a soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\text{Rm}$ of $M$ satisfies the estimate $|\text{Rm}|\leqslant cS$ for some constant $c$. Moreover, the curvature operator $\text{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\text{Rm}\geqslant -c(\ln (r+1))^{-1/4}$, where $r$ is the distance function to a fixed point in $M$. As an application, we prove that if the scalar curvature converges to zero at infinity, then the soliton must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.

scholarly journals Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary II

2020 ◽  
Vol 28 (1) ◽  
pp. 165-179
Author(s):  
Luca Sabatini
Keyword(s):  
Riemannian Manifold ◽  
Lower Bound ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Injectivity Radius ◽  
Topological Invariants ◽  
Manifolds With Boundary ◽  
Laplacian Spectrum ◽  
Convex Boundary

AbstractWe present some estimate of the Laplacian Spectrum and of Topological Invariants for Riemannian manifold with pinched sectional curvature and with non-empty and non-convex boundary with finite injectivity radius. These estimates do not depend directly on the the lower bound of the boundary injectivity radius but on the bounds of the curvatures of the manifold and its boundary.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

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