## Isoperimetric structure of asymptotically conical manifolds.(English)Zbl 1364.53035

Let $$m\geq 2$$ be an integer and $$(L,g_L)$$ be a connected closed Riemannian manifold of dimension $$m-1$$. If $$(M,g)$$ is an $$m$$-dimensional Riemannian manifold such that there exists a compact set $$K\subset M$$ and a diffeomorphism $$M\setminus K\cong(1,\infty)\times L$$, with $g=dr\otimes dr+r^2g_L+0(1)\qquad \text{as } r\to\infty,\tag{1}$ then we say that $$(M,g)$$ is asymptotically conical with link $$(L,g_L)$$.
If $$k\geq 0$$ is an integer, it is said that the expansion (1) holds in $$C^k$$ if $\sum_{l=0}^kr^l|\nabla^l(g-g_C)|=0(1)\qquad \text{as } r\rightarrow\infty.$ Here $$\nabla$$ is the Levi-Civita connection of the cone $$(C,g_C)$$ where $$C=(0,\infty)\times L$$ and $$g_C=dr\otimes dr+r^2g_L$$, and norms are computed with respect to $$g_C$$. Similarly, given $$\alpha\in (0,1)$$, it is said that the expansion (1) holds in $$C^{k,\alpha}$$ if the weighted $$C^{k,\alpha}$$ norms of $$g-g_C$$ tends to zero as $$r\to\infty$$.
In this paper the authors study the isoperimetric structure of asymptotically conical manifolds $$(M,g)$$ of dimension $$m$$ whose link $$(L, g_L)$$ satisfies the conditions:
(2) $$\mathrm{Ric}_L\geq (m-2)g_L$$;
(3) $$\mathrm{Area}(L,g_L)<\omega_{m-1}$$, where $$\omega_{m-1}$$ is the volume of the unit sphere.
The results of this paper are the following:
Theorem 1. Let $$(M,g)$$ be an asymptotically conical manifold of dimension $$m$$ such that the expansion (1) holds in $$C^{2,\alpha}$$ and with $$\lambda_1(-\triangle_L)>(m-1)$$. There are $$\delta>0$$ and $$V_0>0$$ such that the following hold: Let $$V>V_0$$. Let $$r>1$$ be such that $$\mathcal{L}^m_g(B_r)=V$$. There is exactly one $$u_V\in C^{2,\alpha}(L)$$ with $$-\delta\leq u_V\leq\delta$$, such that $$\Sigma_V=\{(r(1+u_V(x)),x):x\in L\}$$ is a CMC surface that encloses volume $$V$$. Moreover, $$\Sigma_V$$ is volume-preserving, the surfaces $$\{\Sigma_V\}_{V>V_0}$$ form a foliation of the complement of a compact subset of $$M$$, and the $$C^{2,\alpha}$$ norms of $$u_V$$ tend to zero as $$V\to \infty$$.
Theorem 3. Let $$(M,g)$$ be an asymptotically conical manifold of dimension $$m$$ whose link $$(L,g_L)$$ satisfies conditions (2) and (3) and such that the expansion (1) holds in $$C^{1,\alpha}$$. There is $$V_0>0$$ with the following property: For every $$V>V_0$$ there is an isoperimetric region of volume $$V$$. Every such region $$\Omega_V$$ is regular and close to $$B_r$$ where $$r>1$$ is such that $$\mathcal{L}_g^m(B_r)=V$$. If the expansion (1) holds in $$C^{2,\alpha}$$, then $$\partial\Omega_V=\Sigma_V$$, where $$\Sigma_V$$ is as in Theorem 1. In particular, $$\Omega_V$$ is the unique isoperimetric region of volume $$V$$.
For related papers see [G. Huisken and S.-T. Yau, Invent. Math. 124, No. 1–3, 281–311 (1996; Zbl 0858.53071); the second author et al., Invent. Math. 197, No. 3, 663–682 (2014; Zbl 1302.53037); Invent. Math. 194, No. 3, 591–630 (2013; Zbl 1297.49078); J. Differ. Geom. 94, No. 1, 159–186 (2013; Zbl 1269.53071)].

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C12 Foliations (differential geometric aspects)

### Citations:

Zbl 0858.53071; Zbl 1302.53037; Zbl 1297.49078; Zbl 1269.53071
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