On the Łojasiewicz–Simon gradient inequality on submanifolds

The volume-preserving Willmore flow

“…In Section 7, we then prove that blow-ups cannot be compact by using a constrained Lojasiewicz-Simon gradient inequality (cf. [33]), resembling the result in [10] in the case of the Willmore flow. Finally, in Section 8, we combine our blow-up procedure with the powerful removability results in [24] and the classification result in [7] to prove Theorem 1.2.…”

The Willmore flow with prescribed isoperimetric ratio

“…From the assumption H f ≡ const, it follows that ∇I(f ) = 0 and hence ∇ H I(0) = 0. As in [33,Propsition 7.3], we can thus apply [32,Corollary 5.2] to deduce that Theorem 5.6 is satisfied in normal directions, i.e. for the functional W with the constraint I = σ.…”

“…In this section, we will prove the following constrained or refined Lojasiewicz-Simon inequality for the Willmore energy subject to the constraint of fixed isoperimetric ratio, cf. [32].…”

A Li-Yau inequality for the 1-dimensional Willmore energy

“…Even so, this cannot happen, since E satisfies a constrained Lojasiewicz-Simon gradient inequality, cf. [Rup20]. This can be proven using [Rup20, Corollary 5.2], since the energies E and L are analytic and the length is of lower order, see also [RS20,Theorem 4.8] for the analogous argument in the case of clamped curves.…”