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4 Added emphasis on last edit

To my understanding, Proposition 1 in this paper of Pacini, TAMS 2003 gives exactly the proof that you ask for in the Riemannian case; namely, that the volume of orbits: $$vol\colon M\to \mathbb R, \quad vol(x)=\int_{G(x)} i^*(vol_M),$$ where $i\colon G(x)\hookrightarrow M$ is the immersion of the $G$-orbit through $x$ and $vol_M$ is the volume form of $M$, is a continuous function on $M$, vanishing exactly at singular orbits.

More precisely, he proves that:

1. the volume function on regular (i.e., principal or exceptional) orbits $vol\colon M^{reg}\to\mathbb R$ is a smooth function;

2. it has a continuous extension $vol\colon M\to\mathbb R$ that is zero on the singular points $M^{sing}=M\setminus M^{reg}$;

3. $vol^2\colon M\to\mathbb R$ is smooth.

Note that Pacini defines the volume of an orbit not by using the volume of the image, but rather by integrating the pull-back of the volume form. These are the same thing only if the immersion is $1$-to-$1$ (e.g., for principal orbits $G/P$). For an exceptional orbit $G/K$, the immersion is $k$-to-$1$, where $k$ is the number of sheets on the covering by a principal orbit $G/P\to G/K$, so the volume of the image is multiplied by $k$. This correction factor is precisely the cardinality of the fiber, $k=|P/K|$, as pointed out by the OP.

Regarding the second question, adapting the proof to the nondegenerate semi-Riemannian case should be straightforward.

Edit: I recently realized that also the classic paper by Hsiang-Lawson, JDG 1971 (see first few lines of page 7) cites the continuity of the volume function above in $M$ (being zero on singular points) and smoothness in the set of regular points. Although they do not provide an explicit proof, they say it is straightforward from the Slice Theorem. There are also many nice examples following that.

3 Added details, corrected false statement

To my understanding, Proposition 1 in this paper of Pacini, TAMS 2003 gives exactly the proof that you ask for in the Riemannian case; namely, that the volume of orbits: $$vol\colon M\to \mathbb R$$ R, \quad vol(x)=\int_{G(x)} i^*(vol_M),$$where i\colon G(x)\hookrightarrow M is the immersion of the G-orbits (considered as a function on G-orbit through M) x and vol_M is the volume form of M, is a continuous function on M, vanishing exactly at singular orbits. When proving the first point above he seemingly ignores Note that a converging sequence \mathcal O_t Pacini defines the volume of an orbit not by using the volume of the image, but rather by integrating the pull-back of the volume form. These are the same thing only if the immersion is 1-to-1 (e.g., for principal orbits could collapse to G/P). For an exceptional orbit \mathcal O_t\to \mathcal O. However, in this case \mathcal O_t G/K, the immersion is a finite sheeted covering of \mathcal O, so the correction factor needed to make the k-to-1, where d-volume smooth k is exactly the number of sheets of this on the covering , which, as you point outby a principal orbit G/P\to G/K, is the cardinality of so the quotient volume of the exceptional isotropy image is multiplied by the principal isotropy. k. This should account for correction factor is precisely the missing case \mathcal O_t\to\mathcal Ocardinality of the fiber, with \mathcal O exceptional, in k=|P/K|, as pointed out by the proofOP. 2 Added information about missing case in the proof To my understanding, Proposition 1 in this paper of Pacini, TAMS 2003 gives exactly the proof that you ask for in the Riemannian case; namely, that the volume$$vol\colon M\to \mathbb R of $G$-orbits (considered as a function on $M$) is a continuous function vanishing exactly at singular orbits.

More precisely, he proves that:

1. the volume function on regular (i.e., principal or exceptional) orbits $vol\colon M^{reg}\to\mathbb R$ is a smooth function;

2. it has a continuous extension $vol\colon M\to\mathbb R$ that is zero on the singular points $M^{sing}=M\setminus M^{reg}$;

3. $vol^2\colon M\to\mathbb R$ is smooth.

He then studies

When proving the dynamics first point above he seemingly ignores that a converging sequence $\mathcal O_t$ of the gradient flow principal orbits could collapse to an exceptional orbit $\mathcal O_t\to \mathcal O$. However, in this case $\mathcal O_t$ is a finite sheeted covering of $vol^2$ (\mathcal O$, so the correction factor needed to make the$d$-volume smooth is exactly the number of sheets of this covering, which, as you point out, is precisely the mean curvature flow on orbits on cardinality of the quotient of the exceptional isotropy by the principal isotropy. This should account for the missing case$M$).\mathcal O_t\to\mathcal O$, with $\mathcal O$ exceptional, in the proof.

Regarding the second question, adapting the proof to the nondegenerate semi-Riemannian case should be straightforward.

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