Generalized minimizing movements for the varifold Canham-Helfrich flow (Q6566012)

The gradient flow dynamics of the Canham-Helfrich energy\N\[\NE_{\mathrm{CH}} = \int_M \left( \frac{\beta}{2}(H-H_0)^2 + \gamma K \right) \mathrm{d}\mathcal{H}^2,\N\]\Nare studied in the generalized setting of oriented curvature varifolds.\N\NFor the static case, the existence of a minimizer varifold is shown and a lower bound for the Canham-Helfrich energy is obtained in terms of the Willmore energy (corresponding to the case \(H_0\), \(\beta=\frac{1}{2}\), and \(\gamma\)) and the mass of the verfifold.\N\NIn the dynamic, ``evolutionary'' case, the authors consider the incremental functional\N\[\NG_{\mathrm{CH},\tau}(V,V^{n-1}_\tau) = E_{\mathrm{CH}}(V) + \frac{1}{2\tau}W^2_p(V,V^{n-1}_{\tau}),\N\]\Nfor a sequence of varifolds \((V^{n-1}_{\tau})_{n\in\mathbb{N}_0}\) at finite time step \(\tau\) and for the Wasserstein metric \(W_p\). They show that a generalized minimizing movement \(\mathbb{R}^+_0 \ni t \mapsto V(t)\) exists for which \(E_{\mathrm{CH}}(V(t)) \leq E_{\mathrm{CM}}V(0)\) for all \(t \geq0\). It is also shown that the existence of such a movement is still guaranteed if the varifold is constrained to a subvarifold, which implies the preservation of volume and symmetry constraints. It is further shown that solutions varifold always have finite and non-zero extend.\N\NLastly, the authors cover the case of multiply covered surfaces and show that among zero genus varifolds, the \(k\)-times covered sphere is the unique minimizer of \(E_{\mathrm{CH}}\), provided that \(H_0\) is not too large. For non-zero genus, they provide, under a specific constraint on \(\beta\) and \(\gamma\), a Li-Yau type estimate for \(E_{\mathrm{CH}}\) that only depends on the multiplicity and that is independent of the genus. Lastly, it is shown that the multiplicity of varifold is preserved under generalized minimizing movements.