For any n-dimensional compact Riemannian Manifold M with smooth metric g, by employing the heat kernel embedding introduced by Bérard-Besson-Gallot (1994, [<CitationRef CitationID="CR1">1</CitationRef>]), we intrinsically construct a canonical t-family of conformal embeddings <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C_{t,k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>: <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$M\rightarrow \mathbb {R}^{q(t)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> sufficiently small, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q(t)\gg t^{-\frac{n}{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>≫</mo> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, and k as a function of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$O(t^l)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mi>l</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$l\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>l</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> in proper sense. Our approach involves finding all these canonical conformal embeddings, which shows the distinctions from the isometric embeddings introduced by Wang-Zhu (2015, [<CitationRef CitationID="CR9">9</CitationRef>]).

Conformal Embeddings via Heat Kernel

期刊论文
发表日期：

2026年3月19日

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Zhitong Su

摘要

For any n-dimensional compact Riemannian Manifold M with smooth metric g, by employing the heat kernel embedding introduced by Bérard-Besson-Gallot (1994, [1]), we intrinsically construct a canonical t-family of conformal embeddings $C_{t,k}$ $C_{t, k}$ : $M\rightarrow \mathbb {R}^{q(t)}$ $M \to R^{q (t)}$ , with $t>0$ $t > 0$ sufficiently small, $q(t)\gg t^{-\frac{n}{2}}$ $q (t) ≫ t^{- \frac{n}{2}}$ , and k as a function of $O(t^l)$ $O (t^{l})$ with $l\ge 2$ $l \geq 2$ in proper sense. Our approach involves finding all these canonical conformal embeddings, which shows the distinctions from the isometric embeddings introduced by Wang-Zhu (2015, [9]).