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Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure $d\sigma_1$ and $d\sigma_2$. For example two $S^{n-1}$ embedded in $\mathbb{R}^n$. Then whether $d\sigma_1\ast d\sigma_2$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^n$?

I know there is an old result due to Ragozin here:

Ragozin

which deals with analytic submanifolds. So I wonder if his results can be extended to smooth manifolds as well. The only place where the analyticness is used is to show that

$$M_1\times M_2\overset{f}{\rightarrow}\mathbb{R}^n$$

defined by

$$f(x,y)=x+y$$

is an analytic map, so its critical set, not critical value, is of null measure.

And if the answer to my question were negative, would it be helpful to assume $M_1$ and $M_2$ are transversal?

Thank you very much.

Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure $d\sigma_1$ and $d\sigma_2$. For example two $S^{n-1}$ embedded in $\mathbb{R}^n$. Then whether $d\sigma_1\ast d\sigma_2$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^n$?

I know there is an old result due to Ragozin here:

Ragozin

which deals with analytic submanifolds. So I wonder if his results can be extended to smooth manifolds as well. The only place where the analyticness is used is to show that

$$M_1\times M_2\overset{f}{\rightarrow}\mathbb{R}^n$$

defined by

$$f(x,y)=x+y$$

is an analytic map, so its critical set, not critical value, is of null measure.

Thank you very much.

Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure $d\sigma_1$ and $d\sigma_2$. For example two $S^{n-1}$ embedded in $\mathbb{R}^{n-1}$. Then whether $d\sigma_1\ast d\sigma_2$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^n$?

I know there is an old result due to Ragozin here:

Ragozin

which deals with analytic submanifolds. So I wonder if his results can be extended to smooth manifolds as well. The only place where the analyticness is used is to show that

$$M_1\times M_2\overset{f}{\rightarrow}\mathbb{R}^n$$

defined by

$$f(x,y)=x+y$$

is an analytic map, so its critical set, not critical value, is of null measure.

And if the answer to my question were negative, would it be helpful to assume $M_1$ and $M_2$ are transversal?

Thank you very much.

Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure $d\sigma_1$ and $d\sigma_2$. For example two $S^{n-1}$ embedded in $\mathbb{R}^n$. Then whether $d\sigma_1\ast d\sigma_2$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^n$?

I know there is an old result due to Ragozin here:

Ragozin

which deals with analytic submanifolds. So I wonder if his results can be extended to smooth manifolds as well. The only place where the analyticness is used is to show that

$$M_1\times M_2\overset{f}{\rightarrow}\mathbb{R}^n$$

defined by

$$f(x,y)=x+y$$

is an analytic map, so its critical set, not critical value, is of null measure.

And if the answer to my question were negative, would it be helpful to assume $M_1$ and $M_2$ are transversal?

Thank you very much.

Thanks for reading my question. Assume we are given two closed $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure $d\sigma_1$ and $d\sigma_2$. For example two $S^{n-1}$ embedded in $\mathbb{R}^{n-1}$. Then whether $d\sigma_1\ast d\sigma_2$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^n$?

I know there is an old result due to Ragozin here:

Ragozin

which deals with analytic submanifolds. So I wonder if his results can be extended to smooth manifolds as well. The only place where the analyticness is used is to show that

$$M_1\times M_2\overset{f}{\rightarrow}\mathbb{R}^n$$

defined by

$$f(x,y)=x+y$$

is an analytic map, so its critical set, not critical value, is of null measure.

And if the answer to my question were negative, would it be helpful to assume $M_1$ and $M_2$ are transversal?

Thank you very much.

Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure $d\sigma_1$ and $d\sigma_2$. For example two $S^{n-1}$ embedded in $\mathbb{R}^{n-1}$. Then whether $d\sigma_1\ast d\sigma_2$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^n$?

I know there is an old result due to Ragozin here:

Ragozin

which deals with analytic submanifolds. So I wonder if his results can be extended to smooth manifolds as well. The only place where the analyticness is used is to show that

$$M_1\times M_2\overset{f}{\rightarrow}\mathbb{R}^n$$

defined by

$$f(x,y)=x+y$$

is an analytic map, so its critical set, not critical value, is of null measure.

And if the answer to my question were negative, would it be helpful to assume $M_1$ and $M_2$ are transversal?

Thank you very much.