When I read the volume compare theorem (first picture below), I think there should be a theorem about the boundary of ball, liking $$ vol (\partial B_r(p)) \le vol(\partial B_r(\tilde p)) $$ Further, the volume of sub-manifold should can be compared. For example, the second picture can be treated as special case (n=2) of volume compare of sub-manifold. I wonder if there is any research on this, and what is the corresponding paper?
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You are working under assumption of a lower bound of the Ricci curvature for your assertions? – Jfischer Jan 21 '24 at 07:57
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@Jfischer Yes, there must be some bound on the curvature. Maybe, it is Ricci curvature, sectional curvature or k-curvature. The detail is not important. – Enhao Lan Jan 21 '24 at 08:38