On the Morse–Bott property of analytic functions on Banach spaces with Łojasiewicz exponent one half

Abstract

It is a consequence of the Morse–Bott Lemma (see Theorems 2.10 and 2.14) that a $C^2$ Morse–Bott function on an open neighborhood of a critical point in a Banach space obeys a Łojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses (Theorems 1, 2, and Corollary 3) for analytic functions on Banach spaces: If the Łojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse–Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Łojasiewicz gradient inequality for an analytic function on a finite-dimensional vector space (Łojasiewicz in Ensembles Semi-analytiques, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, 1965) and the Morse Lemma (Theorems 4 and 5) for functions on Banach spaces with degenerate critical points that generalize previous versions in the literature, and which we also use to give streamlined proofs of the Łojasiewicz–Simon gradient inequalities for analytic functions on Banach spaces (Theorems 9 and 10).

Appendices

Appendix A: Rate of convergence of a gradient flow for a function obeying a Łojasiewicz gradient inequality

We recall the following enhancement of Huang [66, Theorem 3.4.8].

Theorem A.1

(Convergence rate under the validity of a Łojasiewicz gradient inequality) (See Feehan [44, Theorem 3].) Let ${\mathscr {U}}$ be an open subset of a real Banach space ${\mathscr {X}}$ that is continuously embedded and dense in a Hilbert space ${\mathscr {H}}$. Let ${\mathscr {E}}:{\mathscr {U}}\rightarrow \mathbb {R}$ be an analytic function with gradient map ${\mathscr {E}}':{\mathscr {U}}\rightarrow {\mathscr {H}}$ and $x_\infty \in {\mathscr {U}}$ be a critical point, that is, ${\mathscr {E}}'(x_\infty )=0$. Assume that there are constants $c \in (0,\infty )$, and $\sigma \in (0,1]$, and $\theta \in [1/2,1)$ such that

$$\begin{aligned} \Vert {\mathscr {E}}'(x)\Vert _{{\mathscr {H}}} \ge c|{\mathscr {E}}(x) - {\mathscr {E}}(x_\infty )|^\theta , \quad \text {for all } x \in {\mathscr {U}}_\sigma , \end{aligned}$$

(A.1)

where ${\mathscr {U}}_\sigma := \{x\in {\mathscr {X}}: \Vert x-x_\infty \Vert _{\mathscr {X}}< \sigma \}$. Let $u \in C^\infty ([0,\infty ); {\mathscr {X}})$ be a solution to the gradient system

$$\begin{aligned} \dot{u}(t) = -{\mathscr {E}}'(u(t)), \quad t \in (0,\infty ), \end{aligned}$$

(A.2)

and assume that the orbit $O(u) := \{u(t): t\ge 0\} \subset {\mathscr {X}}$ obeys $O(u) \subset {\mathscr {U}}_\sigma$. Then there exists $u_\infty \in {\mathscr {H}}$ such that

$$\begin{aligned} \Vert u(t) - u_\infty \Vert _{\mathscr {H}}\le \Psi (t), \quad t\ge 0, \end{aligned}$$

(A.3)

where

$$\begin{aligned} \Psi (t) := {\left\{ \begin{array}{ll} \displaystyle \frac{1}{c(1-\theta )}\left( c^2(2\theta -1)t + (\gamma -a)^{1-2\theta }\right) ^{-(1-\theta )/(2\theta -1)}, &{} 1/2< \theta < 1, \\ \displaystyle \frac{2}{c}\sqrt{\gamma -a}\exp (-c^2t/2), &{}\theta = 1/2, \end{array}\right. }\nonumber \\ \end{aligned}$$

(A.4)

and $a, \gamma$ are constants such that $\gamma > a$ and

$$\begin{aligned} a \le {\mathscr {E}}(v) \le \gamma , \quad \text {for all } v \in {\mathscr {U}}. \end{aligned}$$

If in addition u obeys Hypothesis A.2, then $u_\infty \in {\mathscr {X}}$ and

$$\begin{aligned} \Vert u(t+1) - u_\infty \Vert _{\mathscr {X}}\le 2C_1\Psi (t), \quad t\ge 0, \end{aligned}$$

(A.5)

where $C_1 \in [1,\infty )$ is the constant in Hypothesis A.2 for $\delta =1$.

We recall the

Hypothesis A.2

(A priori interior estimate for a trajectory) (See Feehan [44, Hypothesis 2.1].) Let ${\mathscr {X}}$ be a Banach space that is continuously embedded in a Hilbert space ${\mathscr {H}}$. If $\delta \in (0,\infty )$ is a constant, then there is a constant $C_1 = C_1(\delta ) \in [1,\infty )$ with the following significance. If $S, T \in \mathbb {R}$ are constants obeying $S+\delta \le T$ and $u \in C^\infty ([S,T); {\mathscr {X}})$, we say that $\dot{u} \in C^\infty ([S,T); {\mathscr {X}})$ obeys an a priori interior estimate on (0, T] if

$$\begin{aligned} \int _{S+\delta }^T \Vert \dot{u}(t)\Vert _{\mathscr {X}}\,dt \le C_1\int _S^T \Vert \dot{u}(t)\Vert _{\mathscr {H}}\,dt. \end{aligned}$$

(4.6)

In applications, $u \in C^\infty ([S,T); {\mathscr {X}})$ in Hypothesis A.2 will often be a solution to a quasi-linear parabolic partial differential system, from which an a priori estimate (4.6) may be deduced. For example, Hypothesis A.2 is verified by Feehan [44, Lemma 17.12] for a nonlinear evolution equation on a Banach space ${\mathcal {V}}$ of the form (see Caps [26], Henry [60], Pazy [96], Sell and You [106], Tanabe [118, 119] or Yagi [131])

$$\begin{aligned} \frac{du}{dt} + {\mathcal {A}}u = {\mathcal {F}}(t,u(t)), \quad t\ge 0, \quad u(0) = u_0, \end{aligned}$$

(4.7)

where ${\mathcal {A}}$ is a positive, sectorial, unbounded operator on a Banach space ${\mathcal {W}}$ with domain ${\mathcal {V}}^2 \subset {\mathcal {W}}$ and the nonlinearity ${\mathcal {F}}$ has suitable properties.

Results on the rate of convergence of a gradient flow defined by a function obeying a Łojasiewicz gradient inequality in specific examples have been proved earlier—see Simon [108] and Adams and Simon [4] for a restricted class of analytic energy functions arising in geometric analysis and Råde [102, Proposition 7.4] for the Yang-Mills energy function on connections on principal bundles over a closed smooth manifold of dimension two or three. For a recent example, see Carlotto, Chodosh, and Rubinstein [27, Theorem 1] for the Yamabe function on Riemannian metrics over closed smooth manifolds of dimension greater than or equal to three.

Appendix B: Morse–Bott functions and quadratic simple normal crossing functions

We are often asked about the relationship between Morse–Bott functions and quadratic simple normal crossing functions as in (1.5), so we explain the relationship in this section for $\mathbb {K}=\mathbb {R}$; the analogous discussion applies for $\mathbb {K}=\mathbb {C}$.

For an integer $p \ge 1$ and writing $\mathbb {R}^* = \mathbb {R}{\setminus }\{0\}$, we let $\mathbb {R}^{p-1} = (\mathbb {R}^p{\setminus }\{0\})/\mathbb {R}^* = S^{p-1}/\{\pm 1\}$ denote real projective space, so $\mathbb {R}\mathbb {P}^0 \cong \{1\}$ and $\mathbb {R}\mathbb {P}^1 \cong S^1$ while $\mathbb {R}\mathbb {P}^{p-1}$ with $p\ge 3$ is obtained by identifying antipodal points of the sphere $S^{p-1}$.

Definition B.1

(Blowup at a point and exceptional divisor) (See Krantz and Parks [69, Definition 6.2.2].) Let $p\ge 1$ be an integer and W be an open neighborhood of the origin in $\mathbb {R}^p$. The blowup of W at the origin is the set

$$\begin{aligned} \widetilde{W} := \{(x,\ell ) \in W \times \mathbb {R}^{p-1} : x \in \ell \}, \end{aligned}$$

where $\pi : \widetilde{W} \ni (x,\ell ) \mapsto x \in W$ is the blowup map and $E := \pi ^{-1}(0) \subset \widetilde{W}$ is the exceptional divisor.

The set $\widetilde{W}$ is a real analytic manifold and the quotient map $\pi : \widetilde{W} \rightarrow W$ is real analytic and restricts to a real analytic diffeomorphism $\pi : \widetilde{W} {\setminus }E \cong W {\setminus }\{0\}$. By viewing $\mathbb {R}^{p-1} = S^{p-1}/\{\pm 1\}$ and $\mathbb {R}^p {\setminus }\{0\} = \mathbb {R}_+\times S^{p-1}$, we may also write

$$\begin{aligned} \widetilde{W}&= \{(x,\ell ) \in W \times \mathbb {R}^{p-1} : x \in \ell \} \\&= \{(x,[u]) \in W \times S^{p-1}/\{\pm 1\}: x \in \mathbb {R}u\} \\&= \{(x,[u]) \in W \times S^{p-1}/\{\pm 1\}: x = \pm |x| u\} \\&= \{(x,u) \in W \times S^{p-1}: (x,u) \sim (y,v) \text { if } |x|=|y| \text { and } u = \pm v \} \\&= \{(s,u) \in \mathbb {R}\times S^{p-1}: su \in W \text { and } (s,u) \sim (t,v) \text { if } (t,v) = \pm (s,u) \}, \end{aligned}$$

where $[u] = \{\pm u\}$ and, in the last line, the blowup map is $\pi : \widetilde{W} \ni [s,u] \mapsto su \in W$ and

$$\begin{aligned} \pi ^{-1}(0) = \{\{0\} \times S^{p-1}\}/\{\pm 1\} \cong S^{p-1}/\{\pm 1\} = \mathbb {R}^{p-1} \end{aligned}$$

is the exceptional divisor.

If W is an open neighborhood of the origin in $\mathbb {R}^p$ or the half-space $\mathbb {H}^p = \{x\in \mathbb {R}^p:x_p\ge 0\}$, then we could alternatively define the blowup of W at the origin to be the real analytic manifold with boundary,

$$\begin{aligned} \widehat{W} := \{(r,u) \in [0,\infty )\times S^{p-1}: ru \in W\}, \end{aligned}$$

following the usual definition of polar coordinates on $\mathbb {R}^p{\setminus }\{0\}$. The map $\pi : \widehat{W} \ni (r,u) \mapsto ru \in W$ is the blowup map and $\pi ^{-1}(0) = \{0\}\times S^{p-1} \cong S^{p-1}$ is now the exceptional divisor.

Suppose now that $U\subset \mathbb {R}^d$ is an open neighborhood of the origin and $f:U \rightarrow \mathbb {R}$ is a $C^2$ function with $f(0)=0$ and $f'(0)=0$ and that is Morse–Bott at the origin in the sense of Definition 1.5 (1). Thus, after possibly shrinking U, we have that ${{\,\mathrm{Crit}\,}}f$ is a $C^2$ submanifold of U of dimension $c = \dim {\text {Ker}}f''(0)$. Moreover, we may further assume that U is connected and so ${{\,\mathrm{Crit}\,}}f \subset f^{-1}(0)$.

Theorem 2.10 and Remark 2.12 (the Morse–Bott Lemma) imply, after possibly shrinking U, that one can find an neighborhood V of the origin in $\mathbb {R}^d$ and a $C^2$ diffeomorphism¹⁶$\Phi :\mathbb {R}^d \supset V \ni y\mapsto x\in U \subset \mathbb {R}^d$ such that $\Phi (0)=0$ and

$$\begin{aligned} f\circ \Phi (y) = \sum _{i=1}^{p} y_i^2 - \sum _{i=p+1}^{p+n} y_i^2, \quad \text {for all } y \in V. \end{aligned}$$

Note that $p+n = d-c$ and

$$\begin{aligned} {{\,\mathrm{Crit}\,}}f\circ \Phi = V\cap \bigcap _{i=1}^{d-c}\{y_i=0\}. \end{aligned}$$

If $n=0$ and thus $1 \le p = d-c$, we may write $(y_1,\ldots ,y_p) = su$, for $s \in [0,\infty )$ and $u \in S^{p-1} \subset \mathbb {R}^p$, so that

$$\begin{aligned} f\circ \varpi (s,u,y_{p+1},\ldots ,y_d) = s^2, \quad \text {for all } (su,y_{p+1},\ldots ,y_d) \in U, \end{aligned}$$

where we define

$$\begin{aligned} \varpi (s,u,y_{p+1},\ldots ,y_d) := \Phi (su,y_{p+1},\ldots ,y_d). \end{aligned}$$

We see that $\varpi$ gives a $C^2$ map from an open neighborhood V of the origin in $[0,\infty )\times S^{p-1}\times \mathbb {R}^{d-p}$ onto $U\subset \mathbb {R}^d$ such that $\varpi (0)=0$ and

$$\begin{aligned} \varpi (\{s=0\}\cap V) = U\cap \bigcap _{i=1}^p\{y_i=0\} = U\cap \bigcap _{i=1}^{d-c}\{y_i=0\}, \end{aligned}$$

and $\varpi$ is a diffeomorphism from $V{\setminus }\{s=0\}$ onto its image.

Similarly, if $p=0$ and thus $1 \le n = d-c$, we may write $(y_1,\ldots ,y_n) = tv$, for $t \in [0,\infty )$ and $v \in S^{n-1} \subset \mathbb {R}^n$, so that

$$\begin{aligned} f\circ \varpi (t,v,y_{n+1},\ldots ,y_d) = -t^2, \quad \text {for all } (tv,y_{n+1},\ldots ,y_d) \in U, \end{aligned}$$

where we define $\varpi (t,v,y_{n+1},\ldots ,y_d) = \Phi (tv,y_{n+1},\ldots ,y_d)$. We see that $\varpi$ gives a $C^2$ map from an open neighborhood of the origin in $[0,\infty )\times S^{p-1}\times \mathbb {R}^{d-p}$ into $\mathbb {R}^d$ such that $\varpi (0)=0$ and

$$\begin{aligned} \varpi (\{t=0\}\cap V) = U\cap \bigcap _{i=1}^n\{y_i=0\} = U\cap \bigcap _{i=1}^{d-c}\{y_i=0\}, \end{aligned}$$

and $\varpi$ is a diffeomorphism from $V{\setminus }\{t=0\}$ onto its image.

Finally, if $n \ge 1$ and $p \ge 1$, we may write $(y_1,\ldots ,y_p) = su$ and $(y_{p+1},\ldots ,y_{p+n}) = tv$, for $s, t \in [0,\infty )$ and $u \in S^{p-1}$ and $v \in S^{n-1}$, so that

$$\begin{aligned} f\circ \varpi (s,t,u,v,y_{p+n+1},\ldots ,y_d) = s^2-t^2, \quad \text {for all } (su,tv,y_{p+n+1},\ldots ,y_d) \in U, \end{aligned}$$

where we define

$$\begin{aligned} \varpi (s,t,u,v,y_{p+n+1},\ldots ,y_d) := \Phi (su,tv,y_{p+n+1},\ldots ,y_d). \end{aligned}$$

We see that $\varpi$ gives a $C^2$ map from an open neighborhood of the origin in

$$\begin{aligned}{}[0,\infty )\times [0,\infty )\times S^{p-1}\times S^{n-1}\times \mathbb {R}^{d-n-p} \end{aligned}$$

into $\mathbb {R}^d$ such that $\varpi (0)=0$ and

$$\begin{aligned} \varpi (\{s=0\}\cap \{t=0\}\cap W) = V\cap \bigcap _{i=1}^{n+p}\{y_i=0\} = V\cap \bigcap _{i=1}^{d-c}\{y_i=0\}, \end{aligned}$$

after possibly shrinking V and $\varpi$ is a diffeomorphism from $W{\setminus }(\{s=0\}\cup \{t=0\})$ onto its image.

Define a diffeomorphism of $\mathbb {R}^2$ by $(t_1,t_2) \mapsto (s,t) = \varphi (t_1,t_2)$ where $t_1=s+t$ and $t_2=s-t$, so that $s = \frac{1}{2}(t_1+t_2)$ and $t = \frac{1}{2}(t_1-t_2)$. Hence, we obtain

$$\begin{aligned}&f\circ \Pi (t_1,t_2,u,v,y_{p+n+1},\ldots ,y_d) = t_1t_2, \quad \\&\quad \text {for all } (\varphi _1(t_1,t_2)u,\varphi _2(t_1,t_2)v,y_{p+n+1},\ldots ,y_d) \in U, \end{aligned}$$

where we define

$$\begin{aligned} \Pi (t_1,t_2,u,v,y_{p+n+1},\ldots ,y_d) := \Phi (\varphi _1(t_1,t_2)u,\varphi _2(t_1,t_2)v,y_{p+n+1},\ldots ,y_d). \end{aligned}$$

We see that $\Pi$ gives a $C^2$ map from an open neighborhood of the origin in

$$\begin{aligned} \{(t_1,t_2)\in [0,\infty )\times \mathbb {R}: |t_2|\le t_1\} \times S^{p-1}\times S^{n-1}\times \mathbb {R}^{d-n-p} \end{aligned}$$

into $\mathbb {R}^d$ such that $\Pi (0)=0$ and

$$\begin{aligned} \Pi (\{t_1=0\}\cap \{t_2=0\}\cap V) = U\cap \bigcap _{i=1}^{d-c}\{y_i=0\}, \end{aligned}$$

and $\Pi$ is a diffeomorphism from $V{\setminus }(\{t_1=0\}\cup \{t_2=0\})$ onto its image.

In the preceding discussion we could have replaced the roles of the blowups $[0,\infty )\times S^{p-1}$ or $[0,\infty )\times S^{n-1}$ by $(\mathbb {R}\times S^{p-1})/\{\pm 1\}$ or $(\mathbb {R}\times S^{n-1})/\{\pm 1\}$ and the roles of the exceptional divisors, $S^{p-1}$ or $S^{n-1}$ by $\mathbb {R}\mathbb {P}^{p-1}$ or $\mathbb {R}\mathbb {P}^{n-1}$, the only difference being an increase in notational complexity. In summary, we have proved the

Proposition B.2

(Pull-back of a Morse–Bott function to a quadratic simple normal crossing function) Let $d \ge 2$ be an integer, $U\subset \mathbb {R}^d$ be an open neighborhood of the origin, and $f:U \rightarrow \mathbb {R}$ be a $C^2$ function that is Morse–Bott at the origin and obeys $f(0)=0$. Then, after possibly shrinking U, there are an open neighborhood V of the origin in $\mathbb {R}^d$ and a $C^2$ map $\pi :V \rightarrow U$ such that $\pi$ restricts to a diffeomorphism from $V{\setminus }\{y_1=0\}$ or $V{\setminus }(\{y_1=0\}\cup \{y_2=0\})$ onto its image and

$$\begin{aligned} \pi ^*f(y) = \pm y_1^2 \quad \text {or}\quad y_1y_2, \quad \text {for all } y=(y_1,\ldots ,y_d) \in V, \end{aligned}$$

and $\pi ({{\,\mathrm{Crit}\,}}f\circ \pi ) = {{\,\mathrm{Crit}\,}}f$, where ${{\,\mathrm{Crit}\,}}f\circ \pi = \{y_1=0\}\cap V$ or $(\{y_1=y_2=0\})\cap V$.

Appendix C: Integrability and Morse–Bott conditions for the harmonic map energy and the area functions

In Sect. 1.5 we defined the concepts of Jacobi vector, integrable Jacobi vector, and integrable critical point (see Definition 1.17). We noted (see Lemma 1.18) that if a function is Morse–Bott at a critical point, then that critical point is integrable. Theorem 8 has been proved by Simon for a specific class of analytic functions on certain Banach spaces (given by $C^{2,\alpha }$ sections of a Riemannian vector bundle over a closed Riemannian manifold) that includes the harmonic map energy and the area functions. We shall give a proof of a more general version of Theorem 8 elsewhere [45], but we outline here how Theorem 8 may be proved; in addition to the references cited below, we also refer the reader to Simon [110, Sects. 3.11–3.14 and 3.13.16] for further expository details.

Proof

(Outline of proof of Theorem 8) In order to avoid notational conflict with the remainder of this section, we let ${\mathscr {E}}$ denote the analytic function considered in Theorem 8. As in Simon’s proof of his infinite-dimensional version [108, Theorem 3] of the Łojasiewicz gradient inequality, one first applies Lyapunov–Schmidt reduction as in [108, Sect. 2] to the function ${\mathscr {E}}:{\mathscr {U}}\rightarrow \mathbb {K}$. This step yields an analytic embedding $\Psi :{\mathscr {V}}\cap {\mathscr {K}}\rightarrow {\mathscr {X}}$ of the intersection with the kernel ${\mathscr {K}}:={\text {Ker}}{\mathscr {E}}''(x_0)$ and an open neighborhood ${\mathscr {V}}$ of the origin in $\tilde{{\mathscr {X}}}$, together with an analytic function $\Gamma = {\mathscr {E}}\circ \Psi :{\mathscr {V}}\cap {\mathscr {K}}\rightarrow \mathbb {K}$ (see Adams and Simon [4, p. 230], Feehan and Maridakis [49, Lemmas 2.3 and 2.5], Simon [108, pp. 538–539], or Simon [109, Part II, Sect. 6]). By hypothesis, $x_0$ is an integrable critical point in the sense of Definition 1.17 and so, after possibly shrinking ${\mathscr {V}}$, the function $\Gamma$ is constant on ${\mathscr {V}}\cap {\mathscr {K}}$ by Adams and Simon [4, Lemma 1, p. 231].

One can now show that ${\mathscr {E}}'(x)=0$ for all $x \in \Psi ({\mathscr {V}}\cap {\mathscr {K}})$, essentially by reversing our proof of [49, Lemma 2.5] or arguing as in Simon [108, p. 539], and thus

$$\begin{aligned} \Psi ({\mathscr {V}}\cap {\mathscr {K}}) \subseteq {{\,\mathrm{Crit}\,}}{\mathscr {E}}. \end{aligned}$$

By hypothesis, ${\mathscr {E}}''(x_0) \in {\mathscr {L}}({\mathscr {X}},\tilde{{\mathscr {X}}})$ is Fredholm with index zero and thus we have ${\mathscr {X}}={\mathscr {X}}_0\oplus {\mathscr {K}}$ and $\tilde{{\mathscr {X}}}\cong {\mathscr {X}}_0\oplus {\mathscr {K}}$ (see Lemma 2.4 (2)). In particular,

$$\begin{aligned} {\text {Ran}}{\mathscr {E}}''(x_0) + {\mathscr {K}}= \tilde{{\mathscr {X}}} \end{aligned}$$

and so, after possibly shrinking ${\mathscr {U}}$, the analytic gradient map ${\mathscr {E}}':{\mathscr {U}}\rightarrow \tilde{{\mathscr {X}}}$ is transverse to the (linear) submanifold ${\mathscr {K}}\subset \tilde{{\mathscr {X}}}$ and hence the preimage $({\mathscr {E}}')^{-1}({\mathscr {K}})$ is an open analytic submanifold of ${\mathscr {X}}$ by the Preimage Theorem from differential topology [73, Proposition II.2.4]. We thus have inclusions

$$\begin{aligned} \Psi ({\mathscr {V}}\cap {\mathscr {K}}) \subseteq {{\,\mathrm{Crit}\,}}{\mathscr {E}}\subseteq ({\mathscr {E}}')^{-1}({\mathscr {K}}), \end{aligned}$$

noting that ${{\,\mathrm{Crit}\,}}{\mathscr {E}}\equiv ({\mathscr {E}}')^{-1}(0)$. Furthermore, we have

$$\begin{aligned} T_{x_0}\Psi ({\mathscr {V}}\cap {\mathscr {K}}) = {\mathscr {K}}= T_{x_0}({\mathscr {E}}')^{-1}({\mathscr {K}}), \end{aligned}$$

where the first equality follows from the construction of $\Psi$ (see [49, Lemma 2.3]) and the second from the observations below:

$$\begin{aligned} T_{x_0}({\mathscr {E}}')^{-1}({\mathscr {K}})&= ({\mathscr {E}}''(x_0))^{-1}(T_{{\mathscr {E}}'(x_0)}{\mathscr {K}}) = ({\mathscr {E}}''(x_0))^{-1}({\mathscr {K}}) \\&= ({\mathscr {E}}''(x_0))^{-1}(0) \quad \text {(since } \tilde{{\mathscr {X}}}={\text {Ran}}{\mathscr {E}}''(x_0)\oplus {\mathscr {K}}) \\&= {\mathscr {K}}\quad \text {(by definition)}. \end{aligned}$$

Hence, after possibly shrinking ${\mathscr {U}}$ or ${\mathscr {V}}$, we have $\Psi ({\mathscr {V}}\cap {\mathscr {K}}) = ({\mathscr {E}}')^{-1}({\mathscr {K}})$ and consequently

$$\begin{aligned} \Psi ({\mathscr {V}}\cap {\mathscr {K}}) = {{\,\mathrm{Crit}\,}}{\mathscr {E}}= ({\mathscr {E}}')^{-1}({\mathscr {K}}). \end{aligned}$$

In particular, ${{\,\mathrm{Crit}\,}}{\mathscr {E}}$ is an open analytic submanifold of ${\mathscr {X}}$ with tangent space $T_{x_0}{{\,\mathrm{Crit}\,}}{\mathscr {E}}= {\text {Ker}}{\mathscr {E}}''(x_0)$ and so ${\mathscr {E}}$ is Morse–Bott at $x_0$ in the sense of Definition 1.9 (1). $\square$

1.1 Appendix C.1: Integrability and Morse–Bott conditions for the harmonic map energy function

Following Lemaire and Wood [76, Sect. 1], we review the concept of integrability of a Jacobi field along a harmonic map and describe the relation between integrability and the Morse–Bott condition for the harmonic map energy function at a harmonic map. We then list a few examples where integrability is known for harmonic maps.¹⁷

We begin by recalling the second variation of the energy for the harmonic energy function ${\mathscr {E}}$ discussed in Sect. 1.7.2. For a smooth two-parameter variation $f_{t,s}:M\rightarrow N$ of a map $f:M\rightarrow N$ with $f_{0,0} = f$ and $\partial f_{t,s}/\partial t|_{(0,0)} = v$ and $\partial ^2 f_{t,s}/\partial s|_{(0,0)} = w$, the Hessian of ${\mathscr {E}}$ at f is defined by

$$\begin{aligned} {\mathscr {E}}''(f)(v,w) := \left. \frac{\partial ^2{\mathscr {E}}(f_{t,s})}{\partial t\partial s}\right| _{(0,0)}, \end{aligned}$$

where ${\mathscr {E}}$ is as in (1.19). One has

$$\begin{aligned} {\mathscr {E}}''(f)(v,w) = (J_f(v),w)_{L^2(M,g)}, \end{aligned}$$

where

$$\begin{aligned} J_f(v) := \Delta v - {\text {tr}}R^N(df,v)df \end{aligned}$$

is called the Jacobi operator, a self-adjoint linear elliptic differential operator. Here, $\Delta$ denotes the Laplacian induced on $f^{-1}TN$ and the sign conventions for $\Delta$ and the curvature $R^N$ are those of Eells and Lemaire [41].

Let v be a vector field along f, that is, a smooth section of $f^{-1}TN$, where $f:M\rightarrow N$ is a smooth map. Then v is called a Jacobi field (for the energy) if $J_f(v) = 0$. The space of Jacobi fields ${\text {Ker}}J_f$ is finite-dimensional and its dimension is called the (${\mathscr {E}}$)-nullity of f.

Definition C.1

(Integrability of a Jacobi field along a harmonic map) (See Lemaire and Wood [76, Definition 1.2].) A Jacobi field v along a harmonic map $f_0:M\rightarrow N$ is said to be integrable if there is a smooth family of harmonic maps, $f_t:M\rightarrow N$ for $t\in (-\varepsilon ,\varepsilon )$, such that $f_t|_{t=0} = f_0$ and $v = \partial f_t/\partial t|_{t=0}$.

The following result is stated by Kwon in her Ph.D. thesis [72] (directed by Simon); it can be deduced from Theorem 8 by applying, for example, methods of Feehan and Maridakis [51].

Theorem C.2

(Integrability of Jacobi fields and manifolds of harmonic maps) (See Kwon [72, Proposition 4.1].) Let $d\ge 2$ be an integer and $\alpha \in (0,1)$ be a constant. Let (M, g) and (N, h) be closed, smooth Riemannian manifolds, with M of dimension d, and assume that there is a smooth isometric embedding $N\subset \mathbb {R}^n$ for some integer n. If $f_0 \in C^\infty (M;N)$ is a harmonic map, so ${\mathscr {E}}'(f_0)=0$, then the following hold:

• (1) If there is a constant $\delta =\delta (f_0,g,h,n,\alpha ) \in (0,1]$ such that

  $$\begin{aligned} U_{f_0,\delta } := \left\{ f \in C^{2,\alpha }(M;N): \Vert f-f_0\Vert _{C^{2,\alpha }(M;\mathbb {R}^n)} < \delta \text { and } {\mathscr {E}}'(f)=0\right\} \end{aligned}$$

  (C.1)

  is an open smooth manifold with tangent space $T_{f_0}U_{f_0,\delta } = {\text {Ker}}{\mathscr {E}}''(f_0)$ at $f_0$, then every Jacobi vector field in ${\text {Ker}}{\mathscr {E}}''(f_0)$ is integrable.

• (2) If (N, h) is real analytic, the isometric embedding $N\subset \mathbb {R}^n$ is real analytic, and every Jacobi vector field in ${\text {Ker}}{\mathscr {E}}''(f_0)$ is integrable, then there is a constant $\delta =\delta (f_0,g,h,n,\alpha ) \in (0,1]$ such that the set $U_{f_0,\delta }$ in (C.1) is an open smooth manifold with tangent space $T_{f_0}U_{f_0,\delta } = {\text {Ker}}{\mathscr {E}}''(f_0)$ at $f_0$.

It follows that for real-analytic target manifolds, all Jacobi fields along all harmonic maps are integrable if and only if the space of harmonic maps is a manifold whose tangent bundle is given by the Jacobi fields [76, p. 470]. By Definition 1.5, the conclusion of Theorem C.2 (2) is equivalent to the assertion that all Jacobi fields along $f_0$ are integrable if and only if the harmonic map energy function ${\mathscr {E}}$ is Morse–Bott at $f_0$.

For a further discussion of integrability and additional references, see Adams and Simon [4, Sect. 1], Kwon [72, Sect. 4.1], and Simon [109, pp. 270–272].

According to [76, Theorem 1.3], any Jacobi field along a harmonic map from $S^2$ to $\mathbb {C}\mathbb {P}^2$ is integrable, where the two-sphere $S^2$ has its unique conformal structure and the complex projective space $\mathbb {C}\mathbb {P}^2$ has its standard Fubini-Study metric of holomorphic sectional curvature 1; see Crawford [34] for additional results.

From the list of examples provided by Lemaire and Wood [76, p. 471], there are few other examples of families of harmonic maps that are guaranteed to be integrable, with the list including harmonic maps from $S^2$ to $S^2$ but excluding harmonic maps from $S^2$ to $S^3$ or $S^4$ [77].

Fernández [52] has proved that the space ${\mathrm {Harm}}_d(S^2,S^{2n})$ of degree-d harmonic maps from $S^2$ into $S^{2n}$ has dimension $2d+n^2$. However, thus far, integrability for such maps is known only when $n=1$. Bolton and Fernandez [18] provide a nice survey of what is known regarding regularity of ${\mathrm {Harm}}_d(S^2,S^{2n})$: they recall that ${\mathrm {Harm}}_d(S^2,S^2)$ is known to be a smooth manifold, outline a proof that ${\mathrm {Harm}}_d(S^2,S^6)$ is also a smooth manifold, and survey results on the structure of ${\mathrm {Harm}}_d(S^2,S^4)$ and why that space is not a smooth manifold.

1.2 Appendix C.2: Integrability and Morse–Bott conditions for the area function

Suppose that $m,n \ge 1$ and $r\ge 2$ are integers and M is a closed, connected, oriented, smooth manifold of dimension m. We let $C^{r,\alpha }(M;\mathbb {R}^n)$ denote the Banach space of $C^{r,\alpha }$ maps from M into $\mathbb {R}^n$, where $\alpha \in [0,1]$, and let ${{\,\mathrm{Imm}\,}}^{r,\alpha }(M;\mathbb {R}^n) \subset C^{r,\alpha }(M;\mathbb {R}^n)$ denote the open subset of $C^{r,\alpha }$ immersions, and let ${{\,\mathrm{Emb}\,}}^{r,\alpha }(M;\mathbb {R}^n) \subset C^{r,\alpha }(M;\mathbb {R}^n)$ denote the open subset of $C^{r,\alpha }$ embeddings. If $\Phi \in C^{r,\alpha }(M;\mathbb {R}^d)$ is an embedding, then $g_\Phi := \Phi ^*g$ is a Riemannian metric on M while if $\Phi$ is an immersion, then $g_\Phi$ may be singular. We now consider the area or volume function,

$$\begin{aligned} {{\,\mathrm{Imm}\,}}^{r,\alpha }(M;\mathbb {R}^n) \ni \Phi \mapsto {\mathscr {E}}(\Phi ) := {\text {Vol}}(M, g_\Phi ) \in [0,\infty ). \end{aligned}$$

Then $\Phi (M)$ is called a critical immersed submanifold or (as customary) a minimal immersed submanifold if ${\mathscr {E}}'(\Phi )=0$, where

$$\begin{aligned} {\mathscr {E}}'(\Phi )\eta = \left. \frac{d}{dt}{\text {Vol}}(M, g_{\Phi +t\Phi _\eta })\right| _{t=0} \end{aligned}$$

for all vector fields $\eta \in C^{r,\alpha }(TM)$. One can show that

$$\begin{aligned} {\mathscr {E}}'(\Phi )\eta = \left( \eta ,{\mathscr {E}}'(\Phi )\right) _{L^2(M)}, \end{aligned}$$

with an explicit expression for the gradient ${\mathscr {E}}'(\Phi )$ provided by the first variation formula—see Calegari [24, Proposition 2.1], Colding and Minicozzi [30, pp. 154–155], Dajczer and Tojeiro [35, Proposition 3.1], Lawson [75], Schoen [104, Sect. 2.1], or Xin [130, Theorem 1.2.2 and Remark 1.2.5].

An explicit expression for the Hessian ${\mathscr {E}}''(\Phi )$ at a critical point $\Phi$ is provided by the second variation formula—see Calegari [24, Proposition 3.1], Colding and Minicozzi [30, pp. 154–155], Lawson [75], Schoen [104, Sect. 2.1], and Xin [130, Theorem 6.1.1].

More generally, if we replace $\mathbb {R}^n$ in the preceding discussion by a connected, smooth manifold N without boundary, then it is known that $C^r(M;N)$ is a smooth Banach manifold—see Abraham [1], Bruveris [23], Eichhorn [42], Eliasson [43], or Wittmann [129]. (It is highly likely that published proofs of this result extend to show that $C^{r,\alpha }(M;N)$ is a Banach manifold when $\alpha \in [0,1]$ and, furthermore, that $W^{k,p}(M;N)$ is Banach manifold for $k\in \mathbb {N}$ and $p\in [1,\infty )$, at least for $k\ge 2$ and $kp>m$, taking note of the Sobolev Embedding Theorem [5, Theorem 4.12].) We refer to Michor and Mumford [85, Sect. 2.1] for their analysis of these spaces in the $C^\infty$ category.

Recall from Dajczer and Tojeiro [35, Corollary 3.7] or Xin [130, Corollary 1.3.4] that there exists no minimal isometric immersion $\Phi : M^m \rightarrow \mathbb {R}^n$ of a compact Riemannian manifold without boundary. Hence, we restrict our attention to cases where M and N are closed or M and N are complete or M is compact with boundary and N is complete.

One could again derive an analogue of Theorem C.2 (giving the relationship between integrability of Jacobi vector fields and the Morse–Bott property of an immersed minimal submanifold) from Theorem 8 or derive an analogue of Theorem C.2 for immersed minimal submanifolds from prior, more general results of Simon [108, 109] and Adams and Simon [4].

Adams and Simon list examples of minimal submanifolds whose Jacobi vector fields are all integrable as well as examples that have nontrivial Jacobi vector fields that are not integrable [4, pp. 249–252]. See also Allard and Almgren [6, Sect. 6], Nagura [89,90,91], Simons [111], and Smith [114, 115] (via Remark C.3) for related examples.

White [126, 127] has shown that for generic $C^r$ Riemannian metrics on a manifold N, there are no closed, immersed, minimal submanifolds $M\subset N$ with nontrivial Jacobi fields; the case of geodesics, including immersed geodesics, was proved earlier by Abraham [2].

Remark C.3

(On the relationship between harmonic maps and minimal surfaces) It is useful to recall the relationship between harmonic maps f from a closed, smooth Riemann surface $(\Sigma ,g)$ into a closed Riemannian manifold (N, h), and immersed minimal surfaces in (N, h), since that relationship enriches our supply of examples. Chern and Goldberg [28, Proposition 5.1] show that if $\Sigma =S^2$ and f is a harmonic immersion, then f is a minimal immersion. More generally, though they assume $(N,h)=\mathbb {R}^3$ with its standard metric and allow $(\Sigma ,g)$ to be a Riemann surface with boundary, Dierkes, Hildebrandt, and Sauvigny prove [38, Theorem 2.6.1] that a conformal map f is minimal if and only if it is harmonic. According to their [38, Definition 2.6.1], they may replace $\mathbb {R}^3$ by $\mathbb {R}^n$ for any $n\ge 2$ and more generally, by any Riemannian manifold (N, h) of dimension $n\ge 2$. Moore [87, Theorem 4.2.2] proves a similar result, namely that (the image of) a (weakly) conformal harmonic map $f:(\Sigma ,g) \rightarrow (N,h)$ is a minimal surface. By restricting to $\Sigma =S^2$, Moore [87, Proposition 4.2.3] recovers the result of Chern and Goldberg: a harmonic two-sphere, $f:(S^2,g_{\mathrm {round}}) \rightarrow (N,h)$, is automatically weakly conformal and hence a parametrized minimal surface. See also [38, pp. 36, 77, 249–250, and 309–311] and their discussion of Lichtenstein’s Theorem on reparameterizing maps of the disk and [38, pp. 249–250] for the relationship between area and energy integrals and the minimization problem.

Remark C.4

(On the interpretation of mean curvature flow as a gradient flow) While there is a wealth of references on mean curvature flow, relatively few treat it as gradient flow for the area (volume) function, thus making it less accessible to gradient flow methods pioneered by Simon [108, 109]. For interpretations of mean curvature flow as a gradient system, we refer the reader to Bellettini [14, Remark 2.8 and Sect. 2.3], Colding, Minicozzi, and Pedersen [33, Sect. 1], Ilmanen [67], Mantegazza [83, p. 7, second paragraph], Ritoré and Sinestrari [101, Equation (4.3)], Smoczyk [116], and Zaal [132]. Shi and Vorotnikov [107] provide a useful recent reference, with a view to applications. For introductions to mean curvature flow, we refer to Ecker [40], Mantegazza [83], Ritoré and Sinestrari [101].

For applications of the DeTurck trick [37] to convert mean curvature flow to a nonlinear parabolic partial differential equation and establish short-time existence, we refer to Andrews and Baker [7], Baker [12], and Leng, Zhao, and Zhao [78].

As in the case of Ricci flow, the interpretation of mean curvature flow as a gradient system can lead to the introduction of a time-varying family of Hilbert spaces—a family of $L^2$ spaces defined by a measure that depends on the time-varying family of immersions [83, Sect. 1.2, page 7].