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Can we find a complete connected noncompact Riemannian manifold $(M^n,g)$ such that the curvature operator $Rm>0$ and $$ \inf_{p \in M} \text{Vol}_gB(p,1)=0? $$

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    $\begingroup$ You should add connected otherwise there are obvious counter-examples. $\endgroup$ Feb 19, 2020 at 16:52
  • $\begingroup$ @MoisheKohan right. $\endgroup$
    – Totoro
    Feb 19, 2020 at 17:18
  • $\begingroup$ I wonder, are there such examples if we only require the sectional curvature to be positive? $\endgroup$ Feb 21, 2020 at 7:40
  • $\begingroup$ You should take a look to Example 1 of the paper "VOLUMES OF SMALL BALLS ON OPEN MANIFOLDS: LOWER BOUNDS AND EXAMPLES" by Croke and Karcher (you can find it here ams.org/journals/tran/1988-309-02/S0002-9947-1988-0961611-7/…), it seems to answer your question positively with an explicit construction. However (as proved in their Theorem 1) the answer is negative for $2$ dimensional manifolds. $\endgroup$
    – Dario
    Feb 22, 2020 at 9:52
  • $\begingroup$ No, they attach the cones to a paraboloid and then smooth. The resulting hypersurface is connected. I’ve tried to detail more in my answer below. $\endgroup$
    – Dario
    Feb 23, 2020 at 8:47

1 Answer 1

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The answer is negative if $\dim M=2$ and positive otherwise, as shown in the paper:

Croke, C. B., & Karcher, H. (1988). VOLUMES OF SMALL BALLS ON OPEN MANIFOLDS: LOWER BOUNDS AND EXAMPLES. AMERICAN MATHEMATICAL SOCIETY (Vol. 309). https://www.ams.org/journals/tran/1988-309-02/S0002-9947-1988-0961611-7/S0002-9947-1988-0961611-7.pdf

The negative result is an immediate consequence of Theorem A, while the positive one follows from Example 1. Their construction for the latter is as follows:

Consider the $d$-dimensional hypersurface given by the paraboloid $\mathcal P = \{x\in \mathbb R^{d+1}\mid x_{d+1}=x_1^2+\ldots+x_d^2\}$. At any point $A$ of $\mathcal P$ one can consider its tangential cone $\mathcal C$, i.e., the Euclidean cone of vertex $V$ with axis along the line $AV$ and that intersects $\mathcal P$ tangentially. One then can consider $\tilde {\mathcal P}$ to be $\mathcal P$ with $\mathcal C$ "attached" (see the Figure at p.765).

This is a piecewise $C^\infty$ hypersurface, with a conical singularity at $V$ and that is $C^1$ at the intersection $\mathcal P\cap \mathcal C$. Outside of these regions, the curvature of $\tilde {\mathcal P}$ is $K>0$, since the curvature of $\mathcal P$ is positive and the one of the cone is zero. Moreover, it is always possible to smooth a conical singularity preserving the $K>0$ bound, and the same is true for the $\mathcal P\cap \mathcal C$ part. (You can maybe look at this thesis, e.g., Lemma 3.2.4.) So we have constructed a smooth hypersurface with $K>0$, and where we can estimate the volume of balls contained in the smoothed conical region by the volumes of the corresponding regions in the non-smoothed cones. Moreover, this operation can be done with a sequence of tangent cones $\mathcal C_n$, as soon as they are sufficiently distant one from the other.

The estimation of the volumes of the balls on the tangent cones is the object of the claim at the end of p.765. In particular, there the authors show that if the vertex of the cone is $V = (k+1,\ldots, 2k+1)$ for $k\ge 5$ (I'm fixing $r=1$), this volume is bounded by the volume of a "spherical spindle" : $$ \operatorname{vol}(B(V,1))\le C_d\sin^{d-2}\theta , $$ where $\theta$ is an angle bounded by $\tan^2\theta\le 6/k$. (Here I chose $\varepsilon =1\le k^2/(2k+2)$.)

This shows that it is possible to attach to $\mathcal P$ a sequence of cones $\mathcal C_n$ with vertices $V_n\to +\infty$, and smooth them out so that $$ \lim_{n\to +\infty}\operatorname{vol}(B(V_n,1))\le (C_d+1) \lim_{n\to +\infty} \sin^{d-2}\left(\theta_n\right) = 0. $$ This proves that the obtained hypersurface (which has positive curvature) satisfies your requirement.

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    $\begingroup$ That's a really cool example! In the nutshell the idea seems to be the following. If we take a round sphere $S^{d-1}$ in $\mathbb R^d$, take a point $p$ far from it and construct the cone with the centre in $p$ that is tangent to $S^{d-1}$, the unit ball centred at $p$ in the surface of the cone will have a very small volume, provided $d>2$. If $d=2$, then this will be a segment of length $2$, of course. That's why the example doesn't work in dimension $2$ :). $\endgroup$ Feb 23, 2020 at 9:44
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    $\begingroup$ Exactly! Very good explanation of the idea! :) $\endgroup$
    – Dario
    Feb 23, 2020 at 9:56

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