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This question is about a variation on the Günther (also known as Günther-Bishop) inequality for manifolds of sectional curvature bounded from above. With Greg Kuperberg, we would deduce from it a conjectured isoperimetric inequality in dimension $4$ which for the moment we only get for small enough domains; see our paper for more details and below for a short account.

Notation and the precise question

Let $(M,g)$ be a Riemannian manifold of dimension $n$, and let us call "candle function" at $x\in M$ the Jacobian of its exponential map, normalized as the function $j_x:Tx M \to \mathbb{R}^+$ such that $$ \mathrm{d}y = j_x(tu) \,\mathrm{d}t \,\mathrm{d}u \quad\mbox{when }y=\exp_x(tu)$$ where $\mathrm{d}y$ is the Riemannian volume, $\mathrm{d}t$ is Lebesgue measure, and $\mathrm{d}u$ is the Riemannian volume of the unit ball of $T_x M$.

Fix a geodesic segment $\gamma$ of length $\ell$, parametrized by arc length, without conjugate points, and consider the function of two parameters $r\le t$ given by $$j(r,t) = j_{\gamma(r)}\big((t-r)\gamma'(r)\big)$$ that is, $j(r,t)$ represent how large the volume element at $\gamma(t)$ looks in the exponential chart based at $\gamma(r)$.

For example, if we denote by $s$ be the function $j$ when $M$ is the hyperbolic space, we get $$ s(r,t) = \sinh^{n-1}(t-r).$$

The question is the following:

Is it true that when $M$ has sectional curvature bounded above by $-1$, then (for all geodesic as above) the expression $$ j(0,\ell) - (n-1) \int_0^\ell j(0,t) \,\mathrm{d}t - (n-1) \int_0^\ell j(r,\ell) \,\mathrm{d}t +(n-1)^2 \int_0^\ell \int_0^t j(r,t) \,\mathrm{d}r \,\mathrm{d}t$$ is minimized when $j=s$, i.e. when the curvature is equal to $-1$ along the geodesic?

The $n=4$ version would be enough for our purpose, but I don't see why it would be easier than the general case.

We conjecture that a stronger inequality, obtained by differentiating twice the above, should be true:

Is it true that when $M$ has sectional curvature bounded above by $-1$, then (for all geodesic as above) the expression $$ -\partial^2_{rt} j -(n-1) \partial_t j +(n-1) \partial_r j +(n-1)^2 j$$ is minimized when $j=s$, i.e. when the curvature is equal to $-1$ along the geodesic?

(here for example $\partial^2_{rt}$ means $\frac{\partial^2}{\partial r\partial t}$)

Comparison with Günther, and what we know

A local version of Günther's inequality is that when $M$ has sectional curvature bounded above by $-1$, $$ \partial_t j / j $$ is minimized when $j=s$. It is easy to derive from this that then $$ \partial_t j - (n-1) j$$ is minimized when $j=s$. So in a sense, what we ask is to improve this comparison from the operator $\partial_t -(n-1)$ to the operator $(-\partial_r -(n-1))(\partial_t-(n-1))$.

We are able to answer positively our question when $n=2$, but our proofs do not extend to higher dimension, at least not easily. We are also able to prove the above with weaker constants instead of the $(n-1)$, but then we get a smallness constraint when we apply the inequality to get the isoperimetric comparison.

In our paper with Greg, we prove that if the above inequality is true, then whenever $n=4$, $M$ is simply connected and has sectional curvature bounded above by $-1$, the isoperimetric profile of $M$ is bounded below by the isoperimetric profile of the $4$-dimensional hyperbolic space. A similar comparison is conjectured in all dimension and also when comparing with Euclidean space (with of course the assumption that sectional curvature is non-positive). This conjecture is known to be true when $n=2$ (Weil 1926, ... Aubin 70's ) or $3$ (Kleiner 1992), and for the Euclidean comparison when $n=4$ (Croke 1984). Right now, we only get the $n=4$, curvature less than $-1$ case for domains of volume less than an explicit constant, due to the limited solution we have to the present question. Note that we get a fully satisfying version of this conjecture for comparison to a sphere.

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  • $\begingroup$ the first highlighted question is for the bounty yes? the second just a conjecture? Or do you want a proof of both? Also, in the first Q, the last integral has t as both a limit and a variable - is this a typo? $\endgroup$ – JonMark Perry May 13 '15 at 19:39
  • $\begingroup$ An answer of the first question in the case $n=4$ would get the bounty. There is no typo there, read $\int_0^\ell \Big(\int_0^t j(r,t) \,dr\Big) \,dt$ or equivalently $\iint_{0\le r\le t\le \ell} j(r,t) \,dr\,dt$. $\endgroup$ – Benoît Kloeckner May 13 '15 at 20:20
  • $\begingroup$ what do you mean by s? I can only see an example function, not a technical definition.here ''for example $∂2rt$ means $∂2∂r∂s$) ''- is this a typo ('t' becomes 's'). Thanks. (typo from just before 'Comparison with Günther, and what we know') $\endgroup$ – JonMark Perry May 15 '15 at 5:09
  • $\begingroup$ @JonMarkPerry: the $s$ in the partial derivative was indeed a typo, thank you. The function $s$ is defined just before the first question: it is the candle function of the hyperbolic space (for any geodesic, as the hyperbolic space is two-point homogeneous). $\endgroup$ – Benoît Kloeckner May 15 '15 at 7:02
  • $\begingroup$ Maybe a look at this paper could help you retro.seals.ch/digbib/view?pid=ensmat-001:1992:38::194 $\endgroup$ – BS. May 17 '15 at 15:52

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