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Isoperimetric estimates in low-dimensional Riemannian products
Annals of Global Analysis and Geometry  (IF0.846),  Pub Date : 2021-03-02, DOI: 10.1007/s10455-021-09757-6
Juan Miguel Ruiz, Areli Vázquez Juárez

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).