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Let $(X, g_0)$ be a $n$-dimensional open manifold with finite volume hyperbolic metric. Suppose $(Y, g)$ is another $n$-dimensional manifold and $f: Y\to X$ a proper degree one map. Then by Storm's generlization of Besson-Courtois-Gallot, we know $$ h^n(g)Vol(g)\ge h^n(g_0)Vol(g_0) $$ Hence the minimal volume of $Y$ must great or equal to that of $X$, by Bishop-Gromv volume comparison.

However the reviewer of http://www.ams.org/mathscinet-getitem?mr=1779901 says that Bessières constructed an example that

Theorem: Let $M$ be a compact, connected and orientable manifold of dimension n≥3. Assume that $int(M)$ admits a hyperbolic complete metric $g_0$ of finite volume, such that the boundary ∂M is a union of tori $T^{n−1}$. Then there exists a manifold $N$, compact with boundary, not homeomorphic to M and a proper map $f:N\to M$ of degree 1, such that the restriction to the boundary is a homeomorphism but $minvol(N)≤vol_{g_0}M$. Furthermore, both manifolds have the same simplicial volume.

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    $\begingroup$ You did not asked a question, but I guess you wonder how the two point you mention can be compatible. Isn't it simply that Storm's result is only for complete manifolds without boundary? $\endgroup$
    – Benoît Kloeckner
    Jan 6, 2016 at 9:46
  • $\begingroup$ @BenoîtKloeckner, I see, so Bessières was calculating the minimal volume of a manifold with boundary. Do I understanding correctly that the of the 'expensive' part of the construction is lies one the part where $N$ is NOT homeomorphic to $M$? Since otherwise, obviously chopping off the cusps will take down the volume by a definite positive number. Sorry, I cannot read French paper very clearly. $\endgroup$
    – user17150
    Jan 6, 2016 at 12:38

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