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We solve the isoperimetric problem in the Lens spaces with large fundamental group. Namely, we prove that the isoperimetric surfaces are geodesic spheres or tori of revolution about geodesics. We also show that the isoperimetric problem in L(3,1) and L(3,2) follows from the proof of the Willmore conjecture by Marques and Neves.
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... Examples include space forms (R n , S n , H n , with the flat, round and hyperbolic metrics, respectively), cylinders (S n × R with the product metric, see [21]) and low dimensional products of space forms with one dimensional circles (S n × S 1 , R n × S 1 , H n × S 1 , 2 ≤ n ≤ 7, see [22]). See also some recent results on RP n and space lenses [33]. On the other hand, seemingly simple products like S 2 × R 2 or S 3 × R 2 , with the product metric, have resulted harder to understand than their factors and their explicit isoperimetric profiles are not known. ...
... Now, from direct computations (see figure 6), we have 10.3v 2 3 > (0.867)I (S 5 ,7.5g 5 0 ) (v), for 140 ≤ v ≤ 2389. Hence, we have, for 140 ≤ v ≤ 2389, (33) I (S 2 ×R 3 ,(g 2 0 +dt 2 )) (v) > (0.867)I (S 5 ,7.5g 5 0 ) (v). Finally, we note that I (S 5 ,2.77g 5 0 ) (v) reaches its maximum at v = V ol((S 5 ,7.5g 5 0 )) 2 ≈ 2388.21, while I (S 2 ×R 3 ,(g 2 0 +dt 2 )) (v) is non decreasing, by Lemma 3.1. ...
We estimate explicit lower bounds for the isoperimetric profiles of the Riemannian product of a compact manifold and the Euclidean space with the flat metric, $(M^m\times \mathbb{R}^n,g+g_E)$, $m,n>1$. In particular, we introduce a lower bound for the isoperimetric profile of $M^m\times \mathbb{R}^n$ for regions of large volume and we improve on previous estimates of lower bounds for the isoperimetric profiles of $S^2 \times \mathbb{R}^2$, $S^3 \times \mathbb{R}^2$, $S^2 \times \mathbb{R}^3$. We also discuss some applications of these results in order to improve known lower bounds for the Yamabe invariant of certain manifolds.
... Viana proved the following dichotomy (Lemma 3.3 in [49]). After passing to a subsequence one of the following holds: ...
We show that the number of genus $g$ embedded minimal surfaces in $\mathbb{S}^3$ tends to infinity as $g\rightarrow\infty$. The surfaces we construct resemble doublings of the Clifford torus with curvature blowing up along torus knots as $g\rightarrow\infty$, and arise from a two-parameter min-max scheme in lens spaces. More generally, by stabilizing and flipping Heegaard foliations we produce index at most $2$ minimal surfaces with controlled topological type in arbitrary Riemannian three-manifolds.
... Regarding L(p, q) we mention the recent result [32], which excludes the Lawson surfaces for infinitely many cases. There the author solves the isoperimetric problem for lens spaces with a large fundamental group: either geodesic spheres or tori of revolution about geodesics. ...
... In particular, Corollary 1.2 is also valid for any manifold for which balls are the isoperimetric sets (i.e. they are the unique sets to attain the minimum in the isoperimetric inequality), or with suitable modifications to the statement, on any manifold on which minimizers to (1.1) are unique up to isometry. The former condition is restrictive and only known for a handful of manifolds (see [34] for a survey of the known results), but includes the round sphere and hyperbolic space with standard metrics, which will be studied in [3] in greater detail: Remark 1.3. Corollary 1.2 holds on hyperbolic space and the round sphere, with the barycenter x Ω replaced by the set centers defined in Examples 2.8 and 2.9 respectively. ...
For a domain $\Omega \subset \mathbb{R}^n$ and a small number $\frak{T} > 0$, let \[ \mathcal{E}_0(\Omega) = \lambda_1(\Omega) + {\frak{T}} {\text{tor}}(\Omega) = \inf_{u, w \in H^1_0(\Omega)\setminus \{0\}} \frac{\int |\nabla u|^2}{\int u^2} + {\frak{T}} \int \frac{1}{2} |\nabla w|^2 - w \] be a modification of the first Dirichlet eigenvalue of $\Omega$. It is well-known that over all $\Omega$ with a given volume, the only sets attaining the infimum of $\mathcal{E}_0$ are balls $B_R$; this is the Faber-Krahn inequality. The main result of this paper is that, if for all $\Omega$ with the same volume and barycenter as $B_R$ and whose boundaries are parametrized as small $C^2$ normal graphs over $\partial B_R$ with bounded $C^2$ norm, \[ \int |u_{\Omega} - u_{B_R}|^2 + |\Omega \triangle B_R|^2 \leq C [\mathcal{E}_0(\Omega) - \mathcal{E}_0(B_R)] \] (i.e. the Faber-Krahn inequality is linearly stable), then the same is true for any $\Omega$ with the same volume and barycenter as $B_R$ without any smoothness assumptions (i.e. it is nonlinearly stable). Here $u_{\Omega}$ stands for an $L^2$-normalized first Dirichlet eigenfunction of $\Omega$. Related results are shown for Riemannian manifolds. The proof is based on a detailed analysis of some critical perturbations of Bernoulli-type free boundary problems. The topic of when linear stability is valid, as well as some applications, are considered in a companion paper.
... Other examples include cylinders of the type (S n × ℝ, g 0 + dt 2 ) by the work of Pedrosa [12], and for the Riemannian product of a low-dimensional space form with S 1 , i.e., (S 1 × ℝ n , dt 2 + g E ) , (S 1 × S n , dt 2 + g 0 ) , (S 1 × ℍ n , dt 2 + g H ) ( 2 ≤ n ≤ 7 ), by the work of Pedrosa and Ritoré [13]. See also the recent results of C. Viana on ℝℙ n and on lens spaces with large fundamental groups [20,21]. Other results in this direction include lower bounds for isoperimetric profiles or characterizations of isoperimetric regions, see, for example, [9,11,14,15,16]. ...
Let \((T^k,h_k)=(S_{r_1}^1\times S_{r_2}^1 \times \cdots \times S_{r_k}^1, dt_1^2+dt_2^2+\cdots +dt_k^2)\) be flat tori, \(r_k\ge \cdots \ge r_2\ge r_1>0\) and \(({\mathbb {R}}^n,g_E)\) the Euclidean space with the flat metric. We compute the isoperimetric profile of \((T^2\times {\mathbb {R}}^n, h_2+g_E)\), \(2\le n\le 5\), for small and big values of the volume. These computations give explicit lower bounds for the isoperimetric profile of \(T^2\times {\mathbb {R}}^n\). We also note that similar estimates for \((T^k\times {\mathbb {R}}^n, h_k+g_E)\), \(2\le k\le 5\), \(2\le n\le 7-k\), may be computed, provided estimates for \((T^{k-1}\times {\mathbb {R}}^{n}, h_{k-1}+g_E)\) exist. We compute this explicitly for \(k=3\). We use symmetrization techniques for product manifolds, based on work of Ros (Global theory of minimal surfaces (Proc. Clay Mathematics Institute Summer School, 2001). American Mathematical Society, Providence, 2005) and Morgan (Ann Glob Anal Geom 30:73–79, 2006).
... Notice, in particular, that this formula for volume extends beyond the injectivity radius π n of L(n; 1), in contrast to most results about volumes of balls in Riemannian manifolds (e.g., [14]). In addition to the potential applications of these ideas to data problems, this seems to be a novel result to add to existing knowledge about the geometry and topology of lens spaces [2,9,15,16,21,22,23,24,28]. ...
... An explicit orthonormal eigenbasis for the Laplacian is given in [16]. Moreover, the isoperimetric problem has been solved in all lens spaces L(n; m) with n large enough [28]. ...
Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In this paper, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the $k$th moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces.
... The problem is also solved outside an explicit compact subset in the moduli space of flat T 2 × R, see [13,29]. The 3-dimensional spherical space forms with large fundamental group were studied by this author in [35]. Finally, we mention the works [16,22] on the description of small isoperimetric regions in general manifolds and [9,11] on the uniqueness of large isoperimetric regions in asymptotic flat manifolds with non-negative scalar curvature and positive mass. ...
We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k\subset \mathbb{RP}^n$ (starting with points). This confirms a conjecture of Burago and Zalgaller from 1988 and extends to higher dimensions previous result of M. Ritor\'{e} and A. Ros on $\mathbb{RP}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in $\mathbb{S}^n$.
... Proof. See Section 3 in [70]. ...
The research carried out in this thesis concerns two important class of stationary surfaces in Differential Geometry, namely isoperimetric surfaces and index one minimal surfaces. The former are solutions of the so called isoperimetric problem, which is to determine the regions of least perimeter among regions of same volume in a given manifold. The latter are critical points of the area functional with Morse index one, i.e., minimal surfaces which admits only one direction where the surface can be deformed so to decrease its area. These are usually constructed via mountain pass arguments. This work focus on the study of these objects when the ambient space is a 3-dimensional spherical space forms, i.e., space form with positive curvature. Our main results classify, at the level of topology, such stationary surfaces in the spherical space forms with large fundamental group. Our first result proves that the solutions of the isoperimetric problem in spherical space forms with large fundamental group are either spheres or tori. It was previously known that solutions with genus zero and one are respectively totally umbilical and flat. Combining our result and this geometric description, we derive that the solutions of the isoperimetric problem are either geodesic spheres or quotients of Clifford tori. Our second result proves that orientable minimal surfaces with index one in the aforementioned spherical space forms have genus at most two. This is a sharp estimate as one can use the continuous one-parameter min-max theory to construct in every 3-dimensional spherical space form an index one minimal surface with genus equal the Heegaard genus of such space which is known to be at most two. Our result confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.
... The list of 3-manifolds where the Conjecture 1.1 is verified is small. In the case of spherical space forms, the only examples are the sphere S 3 , the projective space RP 3 , and the lens spaces L(3, 1) and L(3, 2) [25,33]. The conjecture has also been proved on sufficiently pinched convex hypersurfaces in R 4 , see [1,Section 5]. ...
... Proof. See Section 3 in [33]. ...
We prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two. This confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.
