Existence of minimizing Willmore surfaces of prescribed genus

By Matthias Bauer, Ernst Kuwert. 2003

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In Applied Mathematics

Citations in the corpus

The Willmore Conjecture (2014, background): “This is a fast paced area with many developments currently taking place and so, instead of listing them exhaustively, we point the reader to the survey article on the Willmore functional of Kuwert and Schätzle [30], and to the lecture notes of Rivière [53] where an introduction to the analysis of ”
Minimizers of the Willmore functional under fixed conformal class (2010, unknown): “mersion,π(f∗geuc) = τ}. ✷ We see Mp,n(π(g)) = W(Σ,g,n). Next, infτMp,n(τ) = βnp for the inﬁmum under ﬁxed genus deﬁned in (1.3), and, as the minimum is attained and 4π<βn p <8π, see [Sim93] and [BaKu03], 4π<min τ∈T Mp,n(τ) = βn p <8π. In the following proposition, we consider a slightly ”
Variational Principles for immersed Surfaces with $L^2$-bounded Second Fundamental Form (2010, background): “Kuwert in [1]: they proved that for an arbitrary given closed oriented surface † there is an immersion into R that minimizes the Willmore energy among all smooth immersions of that surface.”
Min-Max theory and the Willmore conjecture (2012, background): “From (3), we also get (6) sup x∈∂I5 area(Φ(x)) = 4π. Informally, the min-max family Φ can be thought of as an element of the relative homotopy group π5(S,G), where S denotes the space of 2-surfaces in S3 as before and G denotes the space of geodesic spheres.”
Immersions with Bounded Second Fundamental Form (2012, background): “Note that the projection onto the normal space only depends on Df (more precisely π⊥ = Id−DfG−1Df , see for instance [4], p.”
$W^{2,2}$-conformal immersions of a closed Riemann surface into $\R^n$ (2010, unknown): “This inequality was confirmed later in [2].”
Lipschitz conformal immersions from degenerating Riemann surfaces with L2-bounded second fundamental forms (2013, unknown): “In [1] it is proved that for g 2 ˇ g < ! m g : (1.”
Energy quantization for Willmore surfaces and applications (2011, methodology): “Following the convention in [Sim93], we introduce and In [BK03], it is proved that for g ≥ 2, there holds ]).”
The Wulff shape minimizes an anisotropic Willmore functional (2004, background): “For the construction of minimizers in the classes of fixed genus we refer to [18] in the case of genus one, and to [1] for arbitrary genus.”
Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data (2010, background): “We want to mention in particular Bauer-Kuwert and Simon [BK, Sn] for existence of closed Willmore surfaces of prescribed genus, Kuwert-Schätzle and Leschke-Pedit-Pinkall [KS1, KS2, LPP] for closed Willmore surfaces of fixed conformal class and Rivière [R] for a far reaching regularity result.”