Approximation of weak geodesics and subharmonicity of Mabuchi energy, II: $\varepsilon$-geodesics

Abstract

The purpose of this article is to study the strict convexity of the Mabuchi functional along a ${{\mathcal {C}}}^{1,{{\bar{1}}}}$-geodesic, with the aid of the $\varepsilon$-geodesics. We proved the $L^2$-convergence of the fiberwise volume element of the $\varepsilon$-geodesic. Moreover, the geodesic is proved to be uniformly fiberwise non-degenerate if the Mabuchi functional is $\varepsilon$-affine.

Appendix

In the following, we will provide a different proof of Theorem 3.6. This one is more complicated, but it will give an accurate estimate for the $L^2$-norm of the difference $f_{\ell }- f$ directly from the convergence of the truncated entropies. We expect that this estimate will be useful for some independent interests.

For the beginning, denote ${\kappa }$ by the auxiliary function on the fiber $X_{\tau }$

$$\begin{aligned} {\kappa }(x, A): = \frac{e^{\chi -A}}{\omega ^n} (x). \end{aligned}$$

(7.1)

As before, we omit the sub-index $\varepsilon _{{\ell }}$ in $\chi$ due to the uniform control on their ${{\mathcal {C}}}^{1,{{\bar{1}}}}$-norms. Then the following result holds.

Lemma 7.1

Suppose we have

$$\begin{aligned} \lim _{{\ell }\rightarrow \infty } \int _X {{\mathfrak {f}}}(\varphi _{\varepsilon _{\ell }}) \log {{\mathfrak {f}}}_A (\varphi _{\varepsilon _{\ell }}) = \int _X {{\mathfrak {f}}}(\varphi )\log {{\mathfrak {f}}}_A(\varphi ). \end{aligned}$$

Then there exists a uniform constant C, only depending on the upper bound of $f_{\ell }$ and f, satisfying

$$\begin{aligned} \lim _{{\ell }\rightarrow \infty } \int _X (f_{\ell }- f)^2 \omega ^n \le \left( 2+ \frac{8C}{{\kappa }_0} \right) \max _{X_{{\tau }}} {\kappa }, \end{aligned}$$

(7.2)

for all ${\kappa }$ small enough (or A large enough).

Here ${\kappa }_0$ is the gap of the volume element ${{\mathfrak {f}}}(\varphi )$ (Eq. 3.20), which is a fixed constant. Therefore, Theorem 3.6 directly follows from Lemma 7.1 if we take $A\rightarrow \infty$.

1.1 The maximum function

In order to prove this lemma, the first step is to investigate the following maximum function. Define a function $F: [0, +\infty ) \rightarrow {\mathbb {R}}$ as

$$\begin{aligned} F(x): = x\log x, \end{aligned}$$

and $F(0) = 0$. Then F is a convex continuous function on its domain. In fact, it is smooth in ${\mathbb {R}}^+$, and its first and second derivatives are

$$\begin{aligned} F'(x) = \log x +1;\ \ \ F''(x) = x^{-1} \end{aligned}$$

Moreover, we can truncate F by a small number ${\kappa }>0$ and introduce the following maximum function

$$\begin{aligned} h_{{\kappa }}(x): = \max \{ x\log x, x\log {\kappa }\}. \end{aligned}$$

This is also a convex and continuous function on $[0, +\infty )$, and it is piecewise smooth in this domain. Its first derivative exists everywhere on ${\mathbb {R}}^+$ except at the point $x={\kappa }$, and we have

$$\begin{aligned} h'_{{\kappa }}(x) = \left\{ \begin{array}{rcl} \log {\kappa }, &{} \text{ for } &{} x< {\kappa }\\ 1+ \log x, &{} \text{ for } &{} x > {\kappa }. \\ \end{array}\right. \end{aligned}$$

(7.3)

Furthermore, we can also compute its second derivative on ${\mathbb {R}}^+$ as

$$\begin{aligned} h''_{{\kappa }}(x) = \left\{ \begin{array}{rcl} 0, &{} \text{ for } &{} x< {\kappa }\\ \delta ({\kappa }), &{}\text{ for }&{} x={\kappa }\\ \frac{1}{x}, &{} \text{ for } &{} x > {\kappa }, \\ \end{array}\right. \end{aligned}$$

(7.4)

where $\delta ({\kappa })$ is the Dirac-delta function at the point $x={\kappa }$. We note that the Fundamental Theorem of Calculus is still satisfied for $h_{{\kappa }}, h'_{{\kappa }}, h''_{{\kappa }}$ on the interval [0, 1], since we can take the differentiation in the sense of the generalized derivatives.

Fixing a point x on the fiber, we introduce another variable $t\in [0,1]$ and take

$$\begin{aligned} u_t: = t f_{\ell }+ (1-t) f = at +b, \end{aligned}$$

where $a: = (f_{\ell }- f)(x)$ and $b: = f(x)$. We note that $u_t$ is non-negative and has a uniform upper bound for all t, ${\ell }$ and x, and it is strictly positive for all $t>0$, or $t=0$ and $f(x) >0$. Define a new composition function as

$$\begin{aligned} F_{{\ell }, {\kappa }}(t): = h_{{\kappa }} (u_t). \end{aligned}$$

Observe that this function $F_{{\ell }. {\kappa }}$ is actually linear in t for all $u_t$ small. Moreover, its derivatives can be written as

$$\begin{aligned}{} & {} F'_{{\ell }, {\kappa }}(t) = \left\{ \begin{array}{l} a \log {\kappa }, \ \text{ for }\ at +b < {\kappa }\\ a\log (at +b) +a ,\ \text{ for }\ at+b > {\kappa }; \\ \end{array}\right. \end{aligned}$$

(7.5)

$$\begin{aligned}{} & {} F''_{{\ell }, {\kappa }}(t) = \left\{ \begin{array}{l} 0, \ \text{ for } \ at +b < {\kappa }\\ a \delta (t_0),\ \text{ for }\ at +b ={\kappa }\\ \frac{a^2}{at+b},\ \text{ for }\ at+b > {\kappa }, \\ \end{array}\right. \end{aligned}$$

(7.6)

where $t_0$ is determined by the equation

$$\begin{aligned} at_0 + b = {\kappa }. \end{aligned}$$

In particular, we have $F'_{{\ell }, {\kappa }}(0) = a\log {\kappa }$ if $f(x) < {\kappa }$, and $F'_{{\ell }, {\kappa }}(0) = a\log b + a$ if $f(x) > {\kappa }$. Then the following convergence holds.

Lemma 7.2

For all A large enough, we have

$$\begin{aligned} \lim _{{\ell }\rightarrow + \infty } \int _X F'_{{\ell }, {\kappa }} (0) \omega ^n = 0. \end{aligned}$$

Proof

When the constant A is large enough, we can assume ${\kappa }< {\kappa }_0/2$ on the fiber $X_{{\tau }}$, where ${\kappa }_0$ is the gap of the fiber-wise volume element of ${{\mathcal {G}}}$ defined in Eq. (3.20). Then the fiber can be completely decomposed into two parts as in Eq. (3.19)

$$\begin{aligned} X_{{\tau }} = P_{A,{\tau }} \bigcup P_{{\tau }}^c, \end{aligned}$$

up to a set of measure zero. Recall that the two sets can be re-written as follows

$$\begin{aligned} P_{A, {\tau }} = \{ x\in X_{{\tau }}; \ \ \ f(x) > {\kappa }(x) \}; \ \ \ P^c_{{\tau }}: = \{ x\in X_{\tau }; \ \ \ f(x) =0 \}. \end{aligned}$$

Then we have

$$\begin{aligned} \int _X F'_{{\ell }, {\kappa }} (0)= & {} \int _{P_{A, {\tau }}} F'_{{\ell }, {\kappa }} (0)+ \int _{P_{{\tau }}^c} F'_{{\ell }, {\kappa }} (0) \nonumber \\= & {} \int _{P_{A, {\tau }}} (f_{{\ell }} - f) \log f + \int _{P_{A, {\tau }}} (f_{{\ell }} - f) + \int _{P_{\tau }^c} f_{{\ell }}\log {\kappa }\nonumber \\= & {} \int _{P_{A, {\tau }}} (f_{{\ell }} - f) \log f + \int _{P_{{\tau }}^c} (f_{\ell }-f)\log {\kappa }+ \int _{P_{{\tau }}^c} f \log {\kappa }\nonumber \\{} & {} + \int _X (f_{\ell }- f) - \int _{P^c_{\tau }} (f_{\ell }- f) \nonumber \\= & {} \int _X (f_{\ell }- f) \log \max \{ f, {\kappa }\} + \int _X (f_{\ell }- f) - \int _{P^c_{{\tau }}} f_{\ell }. \end{aligned}$$

(7.7)

The three terms on the RHS of Eq. (7.7) will all converge to zero as ${\ell }\rightarrow +\infty$, since $|| f ||_{L^{\infty }}, || f_{\ell }||_{L^{\infty }}$ are uniformly bounded and $f_{\ell }\rightarrow f$ weakly in $L^p$ for any $p\ge 1$, and then our result follows. $\square$

1.2 The four cases

Next we will apply the Fundamental Theorem of Calculus on the function $F_{{\ell }, {\kappa }}(t)$ and its first derivative, namely, we have

$$\begin{aligned} F_{{\ell }, {\kappa }}(1) - F_{{\ell }, {\kappa }}(0)= & {} h_{{\kappa }}(f_{\ell }) - h_{{\kappa }}(f) \nonumber \\= & {} \int _0^{t_0} F'_{{\ell }, {\kappa }}(t)dt + \int _{t_0}^1 F'_{{\ell }, {\kappa }}(t)dt. \end{aligned}$$

(7.8)

A first observation is that the point $t_0$ may not be in the integration domain above. Suppose the point x is in the subset $P_{{\tau }}^c$, and then we have $b =f(x) = 0$, $a = f_{{\ell }}(x) > 0$. Therefore, the point $t_0$ belongs to the interval (0, 1) if and only if $0< {\kappa }< a$.

Otherwise, we have $t_0 \ge 1$ if $f_{\ell }(x) \le {\kappa }$, but $t_0 \le 0$ is not possible in this case since it means ${\kappa }\le 0$.

On the other hand, suppose the point x is in the subset $P_{A,{\tau }}$. Then we have $b = f(x) > {\kappa }$ and $a = f_{\ell }(x)- f(x)$. Hence $t_0 \in (0,1)$ if and only if $a< 0$ and $b> {\kappa }> a+b = f_{{\ell }}(x)$.

Otherwise, when $a\ge 0$, we have $f_l(x) \ge f(x) > {\kappa }$ for $t_0 \le 0$; or when $a<0$, we have ${\kappa }\le f_{\ell }(x)$ for $t_0 \ge 1$.

In conclusion, we distinguish all situations into the following four cases:

• (i) $x\in P_{A,{\tau }}$, $f_{\ell }(x) \ge f(x) > {\kappa }$ or $f(x) > f_{\ell }(x) \ge {\kappa }$;

• (ii) $x\in P^c_{{\tau }}$, $f(x) = 0$ and $f_{\ell }(x) \le {\kappa }$;

• (iii) $x\in P^c_{{\tau }}$, $f(x)=0$ and $f_{\ell }(x) > {\kappa }$;

• (iv) $x\in P_{A, {\tau }}$, $f_{\ell }(x)< {\kappa }< f(x)$.

We note that these four cases are disjoint from each other. Then we will discuss case by case. For ${\textbf {Case (i)}}$, we note that $u_t > {\kappa }$ and then $h_{{\kappa }}(u_t) = u_t \log u_t$ for all $t\in [0,1]$. Therefore, we can further compute as follows.

$$\begin{aligned} F_{{\ell }, {\kappa }}(1) - F_{{\ell }, {\kappa }}(0)= & {} F'_{{\ell },{\kappa }}(0) + \int _0^1 \int _0^t F_{{\ell },{\kappa }}''(s) ds dt \nonumber \\= & {} F'_{{\ell },{\kappa }}(0) + \int _0^1 \int _0^t \frac{a^2 ds}{as +b} dt \nonumber \\\ge & {} F'_{{\ell },{\kappa }}(0) + \frac{a^2}{2C}, \end{aligned}$$

(7.9)

where the constant C is the uniform upper bound of $u_t$.

For $\mathbf {Case (ii)}$, we note $u_t\le {\kappa }$ and then $h_{{\kappa }}(u_t) = u_t \log {\kappa }$. Hence we have

$$\begin{aligned} F_{{\ell }, {\kappa }}(1) - F_{{\ell }, {\kappa }}(0)= & {} F'_{{\ell },{\kappa }}(0) + \int _0^1 \int _0^t F_{{\ell },{\kappa }}''(s) ds dt \nonumber \\= & {} F'_{{\ell },{\kappa }}(0). \end{aligned}$$

(7.10)

The two cases above are the easy ones. For the remaining cases, we will utilise Eqs. (7.5) and (7.6) in the computation. In $\mathbf {Case (iii)}$, we note that $h_{{\kappa }}(u_t) = u_t \log {\kappa }$ for $t\le t_0$ and $h_{{\kappa }}(u_t) = u_t \log u_t$ for $t> t_0$. Recall that $t_0 = {\kappa }/a$ in this case, and hence the computation follows.

$$\begin{aligned} F_{{\ell }, {\kappa }}(1) - F_{{\ell }, {\kappa }}(0)= & {} \int _0^{{\kappa }/a} F'_{{\ell }, {\kappa }} (t) dt + \int _{{\kappa }/a}^1 F'_{{\ell },{\kappa }}(t) dt \nonumber \\= & {} \frac{{\kappa }}{a} F'_{{\ell }, {\kappa }} (0) + \int _0^{{\kappa }/a} \int _0^t F''_{{\ell }, {\tau }} (s) ds dt + \int _{{\kappa }/a}^1 F_{{\ell },{\kappa }}'(t) dt \nonumber \\= & {} F'_{{\ell }, {\kappa }} (0) + \int _{{\kappa }/a}^1 \left\{ a + \int _{{\kappa }/a}^t F''_{{\ell }, {\kappa }}(s) ds \right\} dt \nonumber \\= & {} F'_{{\ell }, {\kappa }} (0) + (a - {\kappa }) + \int _{{\kappa }/a}^1 \int _{{\kappa }/a}^t \frac{a^2 ds}{as +b } dt \nonumber \\\ge & {} F'_{{\ell }, {\kappa }} (0) + (a - {\kappa }) + \frac{a^2}{2C} (1- {\kappa }/a)^2 \nonumber \\\ge & {} F'_{{\ell }, {\kappa }} (0) + \frac{a^2}{2C} + a(1- {\kappa }/C) - {\kappa }. \end{aligned}$$

(7.11)

Recall that $a > 0$ in this case, and then we have

$$\begin{aligned} F_{{\ell }, {\kappa }}(1) - F_{{\ell }, {\kappa }}(0) \ge F'_{{\ell }, {\kappa }} (0) + \frac{a^2}{2C} - {\kappa }, \end{aligned}$$

(7.12)

for all ${\kappa }$ small enough. Finally, the most difficult one is $\mathbf {Case (iv)}$. As before, we first note that $h_{{\kappa }}(u_t) = u_t\log u_t$ for $t\le t_0$ and $h_{{\kappa }}(u_t)$ and $h_{{\kappa }} (u_t) = u_t\log {\kappa }$ for $t > t_0$. Then we compute in a similar way.

$$\begin{aligned} F_{{\ell }, {\kappa }}(1) - F_{{\ell }, {\kappa }}(0)= & {} \int _0^{\frac{{\kappa }- b}{a}} F'_{{\ell }, {\kappa }} (t) dt + \int _{\frac{{\kappa }-b}{a}}^1 F'_{{\ell },{\kappa }}(t) dt \nonumber \\= & {} \frac{{\kappa }-b}{a} F'_{{\ell }, {\kappa }} (0) + \int _0^{\frac{{\kappa }-b}{a}} \int _0^t F''_{{\ell }, {\tau }} (s) ds dt + \int _{\frac{{\kappa }-b}{a}}^1 F_{{\ell }, {\kappa }}'(t) dt \nonumber \\= & {} F'_{{\ell }, {\kappa }} (0) + \int _0^{\frac{{\kappa }-b}{a}} \int _0^t \frac{a^2 ds}{as +b} dt + \int ^1_{\frac{{\kappa }-b}{a}} \left\{ \int _0^{\frac{{\kappa }-b}{a}} F''_{{\ell },{\kappa }}(s)ds + a \right\} \nonumber \\\ge & {} F'_{{\ell }, {\kappa }} (0) + a +b + \frac{({\kappa }-b)^2}{2C} + \frac{ ({\kappa }-b)(a+b -{\kappa })}{C} -{\kappa }. \end{aligned}$$

(7.13)

Recall that we have $a+b = f_{\ell }(x) >0$, ${\kappa }-b = {\kappa }- f(x) < 0$ and $a+b - {\kappa }= f_{\ell }(x) - {\kappa }<0$. Hence the following estimate holds.

$$\begin{aligned} F_{{\ell }, {\kappa }}(1) - F_{{\ell }, {\kappa }}(0)\ge & {} F'_{{\ell }, {\kappa }} (0) + \frac{({\kappa }-b)^2 }{2C} - {\kappa }\nonumber \\\ge & {} F'_{{\ell }, {\kappa }} (0) + \frac{ {\kappa }_0^2 }{8C} - {\kappa }. \end{aligned}$$

(7.14)

The last inequality in Eq. (7.14) holds is because that we have picked up ${\kappa }< {\kappa }_0 /2$, and $P_{A, {\tau }}$ is actually the set where $f(x) > 0$ on the fiber which is equal to

$$\begin{aligned} \{ x\in X_{{\tau }}; \ \ \ f (x) > {\kappa }_0 \} \end{aligned}$$

up to a set of measure zero by the gap phenomenon.

Combining with Eqs. (7.9)–(7.14) above, we conclude the following inequality after taking the integral on $X_{{\tau }}$.

$$\begin{aligned} \int _X ( h_{{\kappa }} (f_{\ell }) - h_{{\kappa }} (f) ) \omega ^n\ge & {} \int _X F'_{{\ell }, {\kappa }}(0) \omega ^n - \max _{X_{{\tau }}} {\kappa }\nonumber \\{} & {} + \frac{1}{2C}\int _{\left( P^c_{{\tau }} \bigcap \{ f_{\ell }\le {\kappa }\} \right) ^c \bigcap \left( P_{A,{\tau }} \bigcap \{ f_l< {\kappa }< f \} \right) ^c } (f_{\ell }- f)^2 \omega ^n \nonumber \\{} & {} + \frac{1}{8C} {\kappa }_0^2 \mu \left( P_{A, {\tau }} \bigcap \{ f_l< {\kappa }< f \} \right) . \end{aligned}$$

(7.15)

Then we are ready to prove the main theorem in this section.

Proof

By our choice on the function ${\kappa }$ and the maximum function $h_{{\kappa }}$, it follows

$$\begin{aligned} H_A (\varphi _{\varepsilon _{\ell }}) - H_A (\varphi ) = \int _X ( h_{{\kappa }} (f_{\ell }) - h_{{\kappa }} (f) ) \omega ^n. \end{aligned}$$

(7.16)

Thanks to Corollary 3.5, the LHS of Eq. (7.15) converges to zero as ${\ell }\rightarrow +\infty$. Meanwhile, our Lemma 7.2 implies the first term on the RHS of Eq. (7.15) also converges to zero. Therefore, it implies

$$\begin{aligned} \frac{8C}{{\kappa }_0^2} ( \max _{X_{\tau }} {\kappa }) \ge \lim _{{\ell }\rightarrow +\infty } \mu \left( P_{A, {\tau }} \bigcap \{ f_l< {\kappa }< f \} \right) . \end{aligned}$$

(7.17)

Furthermore, we note that the two subsets $P_{A,{\tau }} \bigcap \{ f_l< {\kappa }< f \}$ and $P^c_{{\tau }} \bigcap \{ f_{\ell }\le {\kappa }\}$ are mutually disjoint. Then the third term on the RHS of Eq. (7.15) can be decomposed into the following three parts.

$$\begin{aligned} \int _X (f_{\ell }-f)^2 - \int _{P^c_{{\tau }} \bigcap \{ f_{\ell }\le {\kappa }\} }(f_{\ell }-f)^2 - \int _{P_{A,{\tau }} \bigcap \{ f_l< {\kappa }< f \} } (f_{\ell }-f)^2. \end{aligned}$$

The first negative term in the equation above can be estimated as

$$\begin{aligned} \int _{P^c_{{\tau }} \bigcap \{ f_{\ell }\le {\kappa }\} }(f_{\ell }-f)^2 \le \max _{X_{\tau }} {\kappa }^2, \end{aligned}$$

(7.18)

and the second negative term can be estimated by Eq. (7.17) as

$$\begin{aligned} \int _{P_{A,{\tau }} \bigcap \{ f_l< {\kappa }< f \} } (f_{\ell }-f)^2 \le C^2 \mu \left( P_{A, {\tau }} \bigcap \{ f_l< {\kappa }< f \} \right) . \end{aligned}$$

(7.19)

Combing with Eqs. (7.16)–(7.20) and take the limit in ${\ell }$, we eventually conclude the following estimate

$$\begin{aligned} \left( 2+ \frac{8C^3}{{\kappa }_0} \right) \max _{X_{\tau }} {\kappa }\ge \lim _{{\ell }\rightarrow +\infty } \int _X (f_{\ell }- f)^2\omega ^n, \end{aligned}$$

(7.20)

and then our result follows. $\square$