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**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)].

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)].

Reviewer: Weihuan Chen (Beijing)