Proof. Consider the involved graphs in polar coordinates $ (x,y)= r(\cos \varphi ,\sin \varphi ) $ . Then, the $ \delta $ -gluing $ \tilde {u}(r,\varphi ):=u_1(r,\varphi ) + \phi (r)\left ( u_2(r,\varphi ) - u_1(r,\varphi ) \right ) $ has the derivatives

$$ \begin{align*} \partial_r \tilde u &= \partial_r u_1 + \phi \cdot \partial_r(u_2-u_1) + \phi' \cdot (u_2-u_1),\\ \partial_\varphi \tilde u &= \partial_\varphi u_1 + \phi \cdot \partial_\varphi(u_2-u_1),\\ \partial_{rr} \tilde u &= \partial_{rr}u_1 + \phi \cdot \partial_{rr}(u_2-u_1) + 2\phi' \cdot \partial_r(u_2-u_1) + \phi" \cdot (u_2-u_1),\\ \partial_{\varphi\varphi} \tilde u &= \partial_{\varphi\varphi} u_1 + \phi \cdot \partial_{\varphi\varphi} (u_2-u_1),\\ \partial_{r \varphi} \tilde u &= \partial_{r \varphi} u_1 + \phi \cdot \partial_{r \varphi} (u_2-u_1) + \phi' \cdot \partial_\varphi (u_2-u_1). \end{align*} $$

Since $ \delta <1 $ , we have $ \left \lvert \phi \right \rvert ,\left \lvert \phi '\right \rvert ,\left \lvert \phi "\right \rvert \leq \frac {M}{\delta ^2} $ . It follows that

(2.1) $$ \begin{align} \left\lVert D^\alpha \tilde{u} - D^\alpha u_1\right\rVert_{C^0(A)} \leq \frac{2M}{\delta^2} \left\lVert u_1 - u_2\right\rVert_{C^2(A)} = \frac{2M}{\delta^2} \left\lVert f_1 - f_2\right\rVert_{C^2(A,\mathbb{R}^3)}, \end{align} $$

for all multiindices $ \alpha $ with $ \left \lvert \alpha \right \rvert \leq 2 $ . Furthermore, with the assumption $ \left \lVert u_i\right \rVert _{C^2(A)}\leq 1$ , we have

(2.2) $$ \begin{align} \left\lVert D^{\alpha}\tilde u\right\rVert_{C^0(A)} \leq \left\lVert D^{\alpha}u_1\right\rVert_{C^0(A)} + \left\lVert D^{\alpha}\tilde u - D^\alpha u_1\right\rVert_{C^0(A)} &\leq 1 + \frac{4 M}{\delta^2}. \end{align} $$

Now, the integrand $ H^2\sqrt {\det G} $ of the Willmore energy, with mean curvature $ H $ and first fundamental form $ G $ , is a differentiable function of the first and second derivatives of $ u $ . Thus, it is Lipschitz on the bounded domain specified by equation (2.2) with some Lipschitz constant $ L(M,\delta )>0 $ . Together with equation (2.1), we obtain

$$ \begin{align*} \tilde H^2 \sqrt{\det \tilde G} \leq H_1^2\sqrt{\det G_1} + L \frac{2M}{\delta^2} \left\lVert f_2 - f_1\right\rVert_{C^2(A,\mathbb{R}^3)}. \end{align*} $$

With the area $ 4\pi \delta $ of the annulus, we conclude

$$ \begin{align*} W(\tilde f) \leq W(f_1) + \frac{8\pi L M}{\delta} \left\lVert f_2 - f_1\right\rVert_{C^2(A,\mathbb{R}^3)}. \end{align*} $$

Proof. The Willmore energy of the catenoid vanishes, and hence the Willmore energy of the catenoid sphere is the sum

$$ \begin{align*} W_{\alpha/\beta}(\lambda,\lambda,\delta) = W^{cap}(\lambda,\delta) + W^{glue}_{\alpha/\beta}(\lambda,\delta), \end{align*} $$

where

$$ \begin{align*} W^{cap}(\lambda,\delta)&=2\pi \left( 1 + \sqrt{1 - \frac{(1+\delta)^2}{\lambda^2}} \right) \end{align*} $$

is the energy of the spherical cap without the gluing region, that is, a round sphere of radius $ \lambda $ with a small cap removed whose boundary circle has radius $ 1+\delta $ . And $ W^{glue}_{\alpha /\beta }(\lambda ,\delta ) $ is the energy of the gluing region (the graph over $ \operatorname {\mathrm {Ann}}[1-\delta , 1+\delta ] $ ). We observe for the cap (recall Definition 2.3)

(2.5) $$ \begin{align} \lim_{\delta\searrow 0} W^{cap}(\lambda,\delta) &= W_{\alpha/\beta}(\lambda,\lambda,0), \end{align} $$

(2.6) $$ \begin{align} \lim_{\lambda\to\infty} W^{cap}(\lambda,\delta) &= 4\pi.\qquad\ \ \qquad \end{align} $$

Using the explicit parametrizations of the sphere and catenoid and the estimations $ \left \lvert \phi \right \rvert \leq 1 $ , $ \left \lvert \phi '\right \rvert \leq \frac {M}{\delta } $ and $ \left \lvert \phi "\right \rvert \leq \frac {M}{\delta ^2}$ of the gluing function, one can show that there exist constants $ \Lambda ,C>0 $ such that for all $ \lambda \geq \Lambda $ and $ \delta \in (0,\frac 12) $ , we have

$$ \begin{align*} W^{glue}_{\alpha}(\lambda,\delta)& \leq C\frac{1}{\delta\lambda^2},\\ W^{glue}_{\beta}(\lambda,\delta)& \leq C \frac{\delta}{\lambda^2},\\ \partial_\lambda W^{glue}_{\beta}(\lambda,\delta) &\geq - C \frac{\delta}{\lambda^3}. \end{align*} $$

See Lemma A.1 for a proof of these estimations. Now, the statements ii) and iii) follow immediately, and to prove i), we set $ \delta _0:=\frac {2\pi }{C} $ to obtain for all $ \lambda \geq \Lambda $ and $ \delta \in (0,\delta _0) $

$$ \begin{align*} \partial_\lambda W^{glue}_{\beta}(\lambda,\delta) &\geq - C \frac{\delta}{\lambda^3} \\ &> - \frac{2\pi}{\lambda^3} \\ &> -\frac{2\pi(1+\delta)^2}{\lambda^2 \sqrt{\lambda^2-(1+\delta)^2}}\\ &= -\partial_\lambda W^{cap}(\lambda,\delta), \end{align*} $$

which yields $ \partial _\lambda W_\beta (\lambda ,\lambda ,\delta )>0 $ and thus statement i).

Proof. Let $ f:\mathbb {S}^2\times [0,T) \to \mathbb {R}^3 $ be the maximal Willmore flow with initial surface $ f_0 $ and $ T\in (0,\infty ] $ . Fix some $ s_0\in (0,T) $ and set

$$ \begin{align*}\omega:=W(f_0)-W(f_{s_0})>0.\end{align*} $$

Let $ A:=\operatorname {\mathrm {Ann}}[1-\delta ,1+\delta ] $ and $\rho>0 $ sufficiently small such that by Lemma 2.2 the gluing excess Willmore energy for a $ \delta $ -gluing between two graphs $ g_1,g_2 \in C^2(A,\mathbb {R}^3) $ with $ \left \lVert g_i\right \rVert _{C^2(A)}\leq 1 \text{ and } \left \lVert g_1 - g_2\right \rVert _{C^2(A,\mathbb {R}^3)}<\rho $ is less than $ \frac {\omega }{5} $ .

Step 1: Prepare catenoids and spheres.

Fix $ \Lambda>0 $ sufficiently large such that for $ \lambda ,R\geq \Lambda $ , catenoids of scale $ \frac {1}{\lambda } $ , rotation axis $ (0,0,1) $ and center $ (0,0,c) $ with $ c\in \left [0,\lambda ^{-1}\text{ arccosh}(\lambda )\right ] $ , and spheres of radius $ R $ and center $ (0,0,m) $ with $ m\in \left [\sqrt {R^2-1},R\right ] $ (see equations (2.3) and (2.4) for $ c=\lambda ^{-1}\text{ arccosh} (\lambda ) $ and $ m=\sqrt {R^2-1} $ ) have graph parametrizations $ u_c, u_s: A \to \mathbb {R} $ over $ A\times \{0\} $ , respectively, which satisfy

(2.9) $$ \begin{align} \left\lVert u_c\right\rVert_{C^2\left(A\right)}<\frac{\rho}{8}, \end{align} $$

(2.10) $$ \begin{align} \left\lVert u_s\right\rVert_{C^2\left(A\right)}<\frac{\rho}{8}. \end{align} $$

Step 2: Fix domains and notation for gluing.

Let $ S_1,S_2\subseteq \mathbb {S}^2 $ be the northern and southern closed hemispheres and fix a closed annulus $ C\subseteq \mathbb {S}^2 $ and two disjoint closed disks $ D_1\subseteq S_1,D_2\subseteq S_2 $ such that they overlap in annuli $ C_i:=D_i \cap C $ with nonempty interior (see Figure 7). To make the gluings of immersions along their graphs over the annulus $ A $ well-defined in the following, we fix once diffeomorphisms $ \psi ^+_i,\psi ^-_i: A \to C_i $ that are orientation preserving and reversing, respectively. We say that an immersion $ h:C_i\to \mathbb {R}^3 $ is a graph over $ A\times \{0\}\subseteq \mathbb {R}^3 $ if $ h\circ \psi ^+_i $ or $ h\circ \psi ^-_i $ is a graph on $ A $ , that is, $ h\circ \psi ^{\pm }(p) = (p,u_h(p)) $ for some $ u_h:A\to \mathbb {R} $ . In the following, before each gluing, we implicitly assume that by orientation-preserving reparametrizations, the involved immersions are graphs over $ A $ in the way just defined. The resulting gluing is always an immersion. Outside of the gluing region, we recover the involved immersions and inside, we obtain a graph, in particular an immersion.

Figure 7

The introduced domains on $ \mathbb {S}^2 $ and the immersion $ f_{t_0} $ .

Step 3: Fix a time right before the singular time.

Fix $ \lambda \geq \Lambda $ and $ \hat f: C \to \mathbb {R}^3 $ that is a parametrization of appropriate orientation of a (precise) catenoid piece of scale $ \frac {1}{\lambda } $ in $ \overline {B_2(0)}\subseteq \mathbb {R}^3 $ such that $ \hat f|_{S_1} $ parametrizes the part in the upper half-space and $ \hat f|_{C_i} $ are graphs over $ A\times \{0\} $ . Then, by Theorem E, there exists $ t_0\in (s_0,T) $ such that, with appropriate reparametrization and rescaling of $ f_{t_0} $ , the error $ \rho _1:=\left \lVert f_{t_0} - \hat {f}\right \rVert _{C^2(C,\mathbb {R}^3)} $ is sufficiently small. We choose it so small that $ f_{t_0}|_{C_i} $ are graphs over $ A\times \{0\} $ (see Figure 7) and their graph parametrizations $ v^i_{t_0} $ of $ f_{t_0}|_{C_i} $ and $ \hat u^i $ of $ \hat f|_{C_i} $ in the upper half-space ( $ i=1 $ ) and in the lower half-space ( $ i=2 $ ) satisfy

(2.11) $$ \begin{align} \left\lVert v^i_{t_0}-\hat u^i\right\rVert_{C^2\left(A\right)}<\frac{\rho}{8}. \end{align} $$

Furthermore, $ \rho _1 $ is chosen such that all $ g:C\to \mathbb {R}^3 $ with $ \left \lVert \hat f - g\right \rVert _{C^2(C,\mathbb {R}^3)}\leq \rho _1 $ satisfy

(2.12) $$ \begin{align} W(g)<W(\hat{f})+\frac{\omega}{5}=\frac{\omega}{5}. \end{align} $$

Step 4: Construct the final homotopy.

We construct three regular homotopies

$$ \begin{align*} H^1&:\mathbb{S}^2\times[0,1]\to\mathbb{R}^3,\\ H^c&:C \times[0,1]\to\mathbb{R}^3,\quad (p,t)\mapsto f_{t_0}(p)+t\left(\hat{f}(p)-f_{t_0}(p)\right),\\ H^2&:\mathbb{S}^2\times[0,1]\to\mathbb{R}^3, \end{align*} $$

where $ H^1,H^2 $ are defined later (see Step 5) and satisfy all of the following (see also Figure 8).

1. i) $ H^i_0|_{D_i}= f_{t_0}|_{D_i} $ ,

2. ii) $ W(H^i_0) < W(f_{t_0}|_{D_i}) + \frac \omega 5, $

3. iii) $ t\mapsto W(H^i_t)$ is nonincreasing,

4. iv) $ H^i_t|_{C_i} $ is a graph over $ A\times \{0\} $ for all $ t\in [0,1] $ ,

5. v) The graph parametrizations $ u^i_t:A\to \mathbb {R} $ of $ H^i_t $ over $ A\times \{0\} $ satisfy

   $$ \begin{align*}\left\lVert u^i_t\right\rVert_{C^2(A)} <\frac{\rho}{2},\end{align*} $$

6. vi) $ H^i_1 $ is a round sphere of radius $ R $ and center $ \left (0,0,\lambda ^{-1}\text{ arccosh}(\lambda ) \pm \sqrt {R^2-1}\right ) $ .

We glue them together to a regular homotopy $ H:\mathbb {S}^2\times [0,1]\to \mathbb {R}^3 $ that for each time $ t\in [0,1] $ is defined on $ D_i $ to be the $ \delta $ -gluing of $ H^i_t|_{D_i} $ with $ H^c_t $ along the graph parametrizations over $ A\times \{0\} $ , for $ i=1,2 $ . And on $ \mathbb {S}^2\setminus (D_1 \cup D_2) $ , it is defined to agree with $ H^c_t $ . The graph parametrizations $ u^{c,i}_t:A\to \mathbb {R} $ of $ H^c_t $ over $ A\times \{0\} $ satisfy with appropriate reparametrizations $ u^{c,i}_t = v^i_{t_0} + t\left (\hat u^i - v^i_{t_0}\right ) $ and thus with equations (2.11) and (2.9)

$$ \begin{align*} \left\lVert u^{c,i}_t\right\rVert_{C^2(A)} &\leq \left\lVert v^i_{t_0} - \hat u^i\right\rVert_{C^2(A)} + \left\lVert\hat u^i\right\rVert_{C^2(A)} + t \left\lVert\hat u^i - v^i_{t_0}\right\rVert_{C^2(A)}< \frac{\rho}{8} + \frac{\rho}{8} + \frac{\rho}{8} < \frac{\rho}{2}. \end{align*} $$

Thus, with v), we have $ \left \lVert u^c_t - u^i_t\right \rVert _{C^2(A,\mathbb {R})} < \rho $ and the gluing excess energy of each of the two $ \delta $ -gluings remains below $ \frac {\omega }{5} $ for all times. Furthermore, by equation (2.12), we have $ W\left (H^C_t\right )< \frac {\omega }{5} $ and obtain with iii) and ii)

$$ \begin{align*} W(H_t)&<W(H^1_t)+W(H^C_t)+W(H^2_t) + \frac{2}{5}\omega\\ &<W(H^1_t)+W(H^2_t)+\frac35\omega\\ &\leq W(H^1_0)+W(H^2_0)+\frac35\omega\\ &< W(f_{t_0})+\omega\\ & < W(f_0). \end{align*} $$

Now, a concatenation of $ f:\mathbb {S}^2\times [0,t_0]\to \mathbb {R}^3 $ with $ H $ is continuous since i) ensures that $ H_0=f_{t_0} $ . We obtain a regular homotopy that ends in $ H_1 $ whose Willmore energy remains below $ W(f_0) $ . Since $ H^c_1 = \hat f $ is a catenoid of the form (2.3) but shifted by $ (0,0,\lambda ^{-1}\text{ arccosh} (\lambda )) $ , we conclude with vi) that $ H_1|_{S_1},H_1|_{S_2}\in \text {CatSph}(\lambda ,R,\delta ) $ .

Figure 8

Simultaneous regular homotopies to a catenoid piece and a round sphere glued to each other at each time resulting in a catenoid sphere of type $ \beta $ (gluing not displayed).

Step 5: It remains to define $ H^i $ , $ i=1,2 $ , which satisfy i)–vi).

We fix $ i=1 $ and hence consider the graphs in the upper half-space of $ \mathbb {R}^3 $ . $ H^2 $ is constructed analogously considering the graphs in the lower half-space. For the purpose of the remaining proof, we shift all the immersions by $ -\left (0,0,\lambda ^{-1}\text{ arccosh} (\lambda )\right ) $ for the catenoid $ \hat f $ to be of the form (2.3) and the sphere obtained in the very end to be of the form (2.4). The regular homotopy $ H^1 $ will be obtained as a concatenation of multiple steps (see also Figure 8)

$$\begin{align*} \begin{array}{rlrl} \mu & \mapsto \mu F_0, & \mu & \in [1, \bar\mu], \\ t & \mapsto \bar\mu F_t, & t & \in [0, 1], \\ \lambda & \mapsto \lambda \bar\mu F_1, & \lambda & \in [1, \bar\lambda], \\ s & \mapsto \bar\lambda \bar\mu F_1 + sy, & s & \in [0,1], \end{array} \end{align*}$$

the details of which will be laid out in the following.

Step 5.1: Construct $ H^1_0 $ by gluing in a flat disk and start the Willmore flow.

Let $ d_1 $ be an appropriate parametrization of the flat disk $ \overline {B_{\frac {1+\delta }{2}}(0)}\times \{0\} $ with graph parametrization $ u_d=0 $ . Let $ H^1_0:\mathbb {S}^2\to \mathbb {R}^3 $ be the gluing of $ d_1 $ with $ f_{t_0}|_{D_1} $ along the graphs over $ \operatorname {\mathrm {Ann}}\left [\frac {1-\delta }{2},\frac {1+\delta }{2}\right ] $ . Appropriate choice of $ \lambda ,\rho _1 $ can ensure that equations (2.9) and (2.11) equally apply to $ \operatorname {\mathrm {Ann}}\left [\frac {1-\delta }{2},\frac {1+\delta }{2}\right ] $ instead of $ A $ . We obtain

(2.13) $$ \begin{align} \left\lVert v^1_{t_0}\right\rVert_{C^2\left(\operatorname{\mathrm{Ann}} \left[\frac{1-\delta}{2},1+\delta\right]\right)} &\leq \left\lVert v^1_{t_0} - \hat u^1\right\rVert_{C^2\left(\operatorname{\mathrm{Ann}} \left[\frac{1-\delta}{2},1+\delta\right]\right)} + \left\lVert\hat u^1\right\rVert_{C^2\left(\operatorname{\mathrm{Ann}} \left[\frac{1-\delta}{2},1+\delta\right]\right)} \notag\\ &< \frac\rho4. \end{align} $$

By doubling the scale, this can be made a standard $ \delta $ -gluing according to Lemma 2.2 and thus the gluing excess energy is less than $ \frac {\omega }{5} $ . We obtain i) and ii). Defining $ H^2_0 $ analogously, these initial surfaces $ H^1_0,H^2_0 $ have Willmore energy

$$ \begin{align*} 4\pi \leq W(H^i_0)< W(f_{t_0}|_{D_i}) + \frac{\omega}{5}, \end{align*} $$

and therefore

$$ \begin{align*} W(H^1_0) &< W(f_{t_0}|_{D_1}) + \frac{\omega}{5}\\ &\leq W(f_{t_0}) - W(f_{t_0}|_{D_2}) + \frac{\omega}{5}\\ &< W(f_{t_0}) - \left( W(H^2_0) -\frac{\omega}{5} \right) + \frac{\omega}{5}\\ &\leq W(f_{t_0}) -4\pi + \frac{2}{5}\omega \\ &< 12\pi - \omega -4\pi +\frac{2}{5}\omega\\ &<8\pi. \end{align*} $$

Therefore, by Theorem A, the Willmore flow $ \mathbb {S}^2\times [0,\infty )\to \mathbb {R}^3 $ with initial surface $ H^1_0 $ converges smoothly to a round sphere. By monotonically rescaling the time interval $ [0,\infty )\to [0,1) $ , we obtain a smooth regular homotopy $ F:\mathbb {S}^2\times [0,1] \to \mathbb {R}^3 $ where we set $ F_1 $ to be the limit round sphere. Now, $ F $ is a regular homotopy of decreasing Willmore energy from $ F_0=H^1_0 $ to a round sphere. In the following, we modify $ F $ only by translation, rotation, scaling and reparametrization. Hence, iii) will remain satisfied.

Step 5.2: Increase scale to ensure low gluing energy.

Let $ p:=d_1^{-1}(0) $ denote the preimage of the center of the flat disk. Then, its tangent space $ T_pF_0 $ is the $ x $ - $ y $ -plane $ \mathbb {R}^2\times \{0\} $ and by translation and rotation of $ F_t $ , we ensure that $ F_t(p)=0 $ and $ {T_pF_t = \mathbb {R}^2\times \{0\} \subseteq \mathbb {R}^3}$ for all $ t\in [0,1] $ . The Willmore flow might converge to a very small sphere or temporarily increase curvature locally, which might violate v). We counteract this by scaling. First, note that for $ r\in (0,1+\delta ) $ , any $ u\in C^2(B_r(0)) $ with $ u(0)=0 $ and $ \nabla u (0) = 0 $ satisfies by Taylor expansion

(2.14) $$ \begin{align} \left\lVert u\right\rVert_{C^2(B_r(0))} \leq \left(\frac{r^2}{2} + r + 1\right)\left\lVert D^2u\right\rVert_{C^0(B_r(0))} \leq 4 \left\lVert D^2u\right\rVert_{C^0(B_r(0))}. \end{align} $$

The norm of the second fundamental form $ (p,t)\mapsto \left \lVert A_t(p)\right \rVert $ of $ F_t $ in $ p\in \mathbb {S}^2 $ assumes a maximum $ A_{\max } $ on the compact spacetime $ \mathbb {S}^2\times [0,1] $ . Therefore, by Lemma A.2, there exists a radius $ r\in (0,1) $ such that for all $ t\in [0,1] $ , the immersion $ F_t $ is locally a graph over $ B_r(0) \subseteq T_pF_t $ with graph parametrization $ u_t $ satisfying

$$ \begin{align*} \left\lVert D^2u_t\right\rVert_{C^0(B_r)} \leq 5 A_{\max}. \end{align*} $$

We now insert the scaling homotopy $ \mu \mapsto \mu F_0 $ , $ \mu \in \left [1,\bar {\mu }\right ], $ where $ \bar \mu> \frac {1+\delta }{r} $ and $ \bar \mu> \frac {40}{\rho }A_{\max } $ . During this scaling, v) is satisfied since equations (2.13) and (2.14) imply for the graph parametrization $ w_\mu (x) = \mu u_0\left (\frac {x}{\mu }\right ) $ of $ \mu F_0 $

$$ \begin{align*} \left\lVert w_\mu\right\rVert_{C^2(A)} &\leq \left\lVert w_\mu\right\rVert_{C^2\left(\overline{B_{1+\delta}(0)}\right)} \\ &\leq 4 \left\lVert D^2 w_\mu\right\rVert_{C^0\left(\overline{B_{1+\delta}(0)}\right)} \\ &=4 \frac1\mu \left\lVert D^2 u_0\right\rVert_{C^0\left(\overline{B_{\frac{1+\delta}{\mu}}(0)}\right)} \\ &\leq 4 \frac1\mu \left\lVert D^2 u_0\right\rVert_{C^0\left(\overline{B_{1+\delta}(0)}\right)} \\ &=4 \frac1\mu \left\lVert D^2 u_0\right\rVert_{C^0\left(\operatorname{\mathrm{Ann}} \left[\frac{1-\delta}{2},1+\delta\right]\right)} \\ &=4 \frac1\mu \left\lVert D^2 v^1_{t_0}\right\rVert_{C^0\left(\operatorname{\mathrm{Ann}} \left[\frac{1-\delta}{2},1+\delta\right]\right)} \\ &<4 \frac1\mu \frac\rho8 < \frac\rho2. \end{align*} $$

Furthermore, we have ensured that after the scaling the graph parametrizations $ z_t(x) = \bar {\mu } u_t\left (\frac {x}{\bar {\mu }}\right ) $ of the scaled Willmore flow $ \bar {\mu } F_t $ over $ A\times \{0\} $ satisfy v), that is,

$$ \begin{align*} \left\lVert z_t\right\rVert_{C^2(A)}&\leq \left\lVert z_t\right\rVert_{C^2\left(\overline{B_{1+\delta}(0)}\right)} \\ &\leq 4 \left\lVert D^2z_t\right\rVert_{C^0\left(\overline{B_{1+\delta}(0)}\right)}\\ &=4 \frac{1}{\bar{\mu}} \left\lVert D^2u_t\right\rVert_{C^0\left(\overline{B_{\frac{1+\delta}{\bar \mu}}(0)}\right)}\\ &\leq 4 \frac{1}{\bar \mu} \left\lVert D^2u_t\right\rVert_{C^0\left(B_r(0)\right)}\\ &\leq 4 \frac{1}{\bar \mu} 5 A_{\max}\\ &< \frac\rho2. \end{align*} $$

Step 5.3: Append final correcting homotopies.

During this regular homotopy, we reparametrize $ \bar \mu F_t $ appropriately such that $ \bar \mu F_t|_{C_i} $ is a graph over $ A\times \{0\} $ for iv) to hold. Finally, we append a scaling regular homotopy that brings the round sphere to any given radius $ R\geq \Lambda $ . It is now tangent to the $ x $ - $ y $ -plane and centered at $ (0,0,R) $ (type $ \beta $ ) or $ (0,0,-R) $ (type $ \alpha $ ). Then, we append a translation that shifts the center of the sphere from $ \pm (0,0,R) $ to $ \pm \left (0,0,\sqrt {R^2-1}\right ) $ . The translation brings the sphere into the form as described in Definition 2.3. In the case of the sphere lying entirely above the horizontal plane (type $ \beta $ ), it is now tangent to the catenoid $ H^c_1 $ in the unit circle $ \mathbb {S}^1 \times \{0\} $ . By equation (2.10), these two steps do not violate v). Reversing the shift of all surfaces by $ -\left (0,0,\lambda ^{-1}\text{ arccosh}(\lambda )\right ) $ , we observe that the sphere satisfies vi).

Proof of Theorem 1.2.

Fix $ \Lambda ,\delta _0 $ from Lemma 2.4 i) and $ \delta \in (0,\delta _0) $ sufficiently small such that $ \varepsilon := 4\pi - W_\beta (\Lambda ,\Lambda ,\delta ) $ is positive (see Lemma 2.4 ii)). Then, choose $ \lambda _1>\Lambda $ sufficiently large such that $ W_{\alpha /\beta }(\lambda ,\lambda ,\delta )< 4\pi + \varepsilon $ for all $ \lambda>\lambda _1 $ (see Lemma 2.4 iii)). Apply Proposition 2.7 for $ \delta $ and $ R=\lambda>\lambda _1 $ . Let $ H $ be the obtained homotopy and denote $ g_0:=H_1 $ and $ g^i_0:=H_1|_{S_i} \in \text {CatSph}(\lambda ,\lambda ,\delta ) $ , $ i=1,2 $ . If $ g^1_0,g^2_0 $ are both catenoid spheres of type $ \alpha $ , then $ g_0\in J $ and we are done. We may now assume that $ g^1_0 $ is a catenoid sphere of type $ \beta $ . We deform $ g^1_0 $ by a homotopy of catenoid spheres $ g^1_t := \lambda _tf_{\lambda _t} $ with $ f_{\lambda _t}\in \text {CatSph}_\beta (\lambda _t,\lambda _t,\delta ) $ and

$$ \begin{align*}\lambda_t = (1-t)\lambda + t\Lambda,\quad t\in[0,1].\end{align*} $$

By Lemma 2.4 and $ \lambda>\Lambda $ and $\delta <\delta _0 $ , this is a regular homotopy of decreasing Willmore energy to an immersion $ g^1_1\in \text {CatSph}_\beta (\Lambda ,\Lambda ,\delta ) $ of Willmore energy

$$ \begin{align*}W(g^1_1) = W_\beta(\Lambda,\Lambda,\delta)=4\pi - \varepsilon.\end{align*} $$

This regular homotopy leaves the center piece of the catenoid fixed at all times and thus naturally extends to a regular homotopy $ g_t $ that is equal to $ g^1_t $ on $ S_1 $ and $ g^2_0 $ on $ S_2 $ . We conclude

$$ \begin{align*}W(g_1) = W(g^1_1)+W(g^2_0) < 4\pi - \varepsilon + 4\pi + \varepsilon = 8\pi.\end{align*} $$

Therefore, by Theorem A, the Willmore flow with initial surface $ g_1 $ yields the last part of the desired homotopy to a round sphere.

Proof. Fix $ \Lambda>0 $ from Lemma 2.4 iii). Then, we choose $ \delta \in \left (0,\frac 12\right ) $ and $ \lambda _1 \geq \Lambda $ such that

(2.15) $$ \begin{align} W(\lambda,\lambda,\delta)<\frac\omega2 \end{align} $$

for all $ \lambda \geq \lambda _1 $ . Let $ j $ be a smoothly immersed sphere such that on each hemisphere $ S_i $ it is a catenoid sphere $ j|_{S_i}\in \text {CatSph}_\alpha (\lambda _1,\lambda _1,\delta ) $ with outer normal on the spherical parts. It has Willmore energy

$$ \begin{align*} W(j) = 2W_\alpha(\lambda_1,\lambda_1,\delta)<\omega. \end{align*} $$

Now, for given $ f\in I(\omega )\, \cap \, J $ , we apply Proposition 2.7 for $ \delta $ and $ \lambda =R>\lambda _1 $ to obtain a regular homotopy to some $ i\in \text {Imm}(\mathbb {S}^2,\mathbb {R}^3) $ with hemispheres $ i|_{S_i}\in \text {CatSph}_\alpha (\lambda ,\lambda ,\delta ) $ . Type $ \beta $ cannot occur by Theorem D. We concatenate the regular homotopy that deforms each hemisphere by $ t\mapsto i_t|_{S_i}\in \text {CatSph}(\lambda -t,\lambda -t,\delta ) $ , $ t\in [0,\lambda -\lambda _1] $ . By equation (2.15), we have $ W(i_t)<\omega $ for all $ t\in [0,1] $ . The obtained surface has either outer or inner normal on the spherical parts and thus can be joined to $ j $ or $ -j $ by reparametrizing homotopies.

Proof. We repeat many arguments of the proof of Proposition 2.7 and Theorem 1.2 without going into detail. If the Willmore flow with initial surface $ f $ does not converge to a round sphere, it develops a singularity whose blow-up is either a catenoid, a trinoid, or Enneper’s surface (see Theorem E). The latter was ruled out by assumption.

Trinoid case:

Analogously to the catenoid case in Proposition 2.7, each of the three disks that close up the trinoid has Willmore energy less than $ 8\pi $ , and with the same gluing procedure, we arrive at a trinoid whose ends are $ \alpha $ -connected or $ \beta $ -connected to round spheres. Note that Lemma 2.4 uses the asymptotics of the catenoid and thus equally applies to catenoid ends of the trinoid. So, if one of the connections at a catenoid end is of type $ \beta $ , we employ Lemma 2.4 as in the proof of Theorem 1.2 to arrive at an immersion of Willmore energy less than $ 12\pi $ from which we apply Proposition 2.7 to obtain the case (a) or (b). But if all are of type $ \alpha $ , we arrive at (d).

Catenoid case:

The disks that close up a catenoid singularity now each have Willmore energy less than $ 12\pi $ , but only one of them can have Willmore energy larger than $ 8\pi $ . For this part, we proceed as in the proof of Proposition 2.7, but instead of using the Willmore flow to a round sphere to construct the homotopy $ H^1 $ , we use the regular homotopy obtained from Proposition 2.7 to one of the cases (a) or (b). In the former case, the remaining proof carries through to yield (a) or (b) for the whole surface, whereas in the latter case we obtain a catenoid piece that is attached to a sphere and to a surface of type (b). Both connections can be of either type $ \alpha $ or type $ \beta $ . If one of them is of type $ \beta $ , we employ Lemma 2.4 to get below an energy of $ 12\pi $ and apply Proposition 2.7 again. But if both connections are of type $ \alpha $ , we arrive at a surface described in (c).