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Volume comparison via boundary distances. (English) Zbl 1230.53042
Bhatia, Rajendra (ed.) et al., Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency (ISBN 978-981-4324-32-8/hbk; 978-81-85931-08-3/hbk; 978-981-4324-30-4/set; 978-981-4324-35-9/ebook). 769-784 (2011).
A compact Riemannian manifold $$\left\langle M,g\right\rangle$$ with boundary is said to be simple if (i) $$\partial M$$ is strictly convex; (ii) every geodesic segment in $$M$$ realizes the distance between its end points; (iii) geodesics in $$M$$ have no conjugate points. A compact orientable Riemannian manifold $$\left\langle M,g\right\rangle$$ with boundary is a minimal filling of its boundary distance function if, for every compact orientable Riemannian manifold $$\left\langle M^{\prime },g^{\prime }\right\rangle$$ such that $$\partial M^{\prime }=\partial M$$ and $$d_{M^{\prime }}( x,y) \geq d_{M}( x,y)$$ for every $$x,y\in \partial M$$, it follows that $$\text{Vol}_{g^{\prime }}( M^{\prime }) \geq \text{Vol}_{g}( M)$$. The author discusses the following unsolved Conjecture I (in the paper, Conjecture 1.6) and its stronger version, Conjecture II (in the paper, Conjecture 1.6$$^{+}$$).
Conjecture I. Every simple manifold is a minimal filling.
Conjecture II. Every simple manifold is a unique minimal filling of its boundary distance function, up to an isometry fixing the boundary.
In particular, the author shows that Conjecture II implies R. Michel’s boundary rigidity conjecture [Invent. Math. 65, 71–83 (1981; Zbl 0471.53030)], Conjecture I implies M. Gromov’s circle filling conjecture [J. Differ. Geom. 18, 1–147 (1983; Zbl 0515.53037)] and would give another proof of the familiar Hopf theorem (i.e., an $$n$$-torus with a Riemannian metric without conjugate points is flat). The main part of the paper deals with two theorems of the author and D. Burago in [Ann. Math. (2) 171, No. 2, 1183–1211 (2010; Zbl 1192.53048); “Area minimizers and boundary rigidity of almost hyperbolic metrics” (in preparation), arXiv:1011.1570] about filling minimality and boundary rigidity.
For the entire collection see [Zbl 1220.00032].

##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
##### Keywords:
filling volume; minimal filling; boundary distance rigidity
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