When I read the paper “Hillel Gauchman, Pinching theorems for totally real minimal sub manifolds of $\mathbb{C}P^n(c)$”, I don't understand why the following function $f_L$ is continuous?
Let $M$ be a Riemannian manifold and $L$ be a $(0,k)$-tensor field on $M$. At any $x\in M$, $L$ can be considered as a multilinear map $L:T_xM\times \cdots \times T_xM\to \mathbb R$. Let $UM$ be the unit tangent bundle of $M$ and $UM_x$ its fiber at $x\in M$. We say that $u \in UM_x$ is a maximal direction at $x$ with respect to $L$, if it satisfies $L(u,\cdots ,u)=\max_{v\in UM_x}L(v,\cdots ,v)$. For any $x\in M$, set $f_L(x)=L(u,\cdots ,u)$, where $u$ is a maximal direction at $x$ with respect to $L$.
Perhaps this is similar to another question, namely, is the maximum value of the section curvature at each point on the Riemannian manifold a continuous function?
If you trivialize $TM$ near $x\in M$, you have a family of functions on $S^{n-1}$, namely $g_x:S^{n-1} \to \mathbb{R}$ via $g_x(v)=L(v,...,v)$ and want to show that the min value of $g_x$ is continuous in $x$. This is covered here.