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- W2794594825 abstract "Let ${g(t)}_{tin [0,T)}$ be the solution of the Ricci flow on a closed Riemannian manifold $M^n$ with $ngeq 3$. Without any assumption, we derive lower volume bounds of the form ${rm Vol}_{g(t)}geq C (T-t)^{frac{n}{2}}$, where $C$ depends only on $n$, $T$ and $g(0)$. In particular, we show that $${rm Vol}_{g(t)} geq e^{ Tlambda-frac{n}{2}} left(frac{4}{(A(lambda-r)+4B)T}right)^{frac{n}{2}}left(T-tright)^{frac{n}{2}},$$ where $r:=inf_{|phi|_2^2=1} int_M Rphi^2 d{rm vol}_{g(0)}$, $lambda:=inf_{|phi|_2^2=1} int_M 4|nablaphi|^2+Rphi^2 d{rm vol}_{g(0)}$ and $A,B$ are Sobolev constants of $(M,g(0))$. This estimate is sharp in the sense that it is achieved by the unit sphere with scalar curvature $R_{g(0)}=n(n-1)$ and $A=frac{4}{n(n-2)}omega_n^{-frac{2}{n}}$, $B=frac{n-1}{n-2}omega_n^{-frac{2}{n}}$. On the other hand, if the diameter satisfies ${rm diam}_{g(t)}leq c_1sqrt{T-t}$ and there exist a point $x_0in M$ such that $R(x_0,t)leq c_2(T-t)^{-1}$, then we have ${rm Vol}_{g(t)}leq C (T-t)^{frac{n}{2}}$ for all $t>frac{T}{2}$, where $C$ depends only on $c_1,c_2,n,T$ and $g(0)$." @default.
- W2794594825 created "2018-04-06" @default.
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- W2794594825 date "2018-03-26" @default.
- W2794594825 modified "2023-09-24" @default.
- W2794594825 title "Volume bounds of the Ricci flow on closed manifolds" @default.
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- W2794594825 doi "https://doi.org/10.48550/arxiv.1803.09591" @default.
- W2794594825 hasPublicationYear "2018" @default.
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