From Boundary Energy to Volume Energy
── The Einstein–Yamagishi Volume Restoration Equation and a Dimensional-Hierarchical Model of the Universe Based on the $n = 2$ Uniqueness Theorem ──
In this paper, we take as our starting point the $n = 2$ uniqueness theorem proven in the author’s previous works (in the 3-dimensional Euclidean space $\R^3$, the unique condition for the shell mass growth rate $dm/dr$ under a spherically symmetric power-law density $\rho(r) = r^{-n}$ to be uniform is $n = 2$, with constant value $4\pi$), and normalize the geometric mass parameter of 3-dimensional space as $m = 4\pi$.
Under this normalization, reformulating $E = mc^2 = 4\pi c^2$ reveals that this is geometrically identical to the surface area of a 2-dimensional sphere $S^2$ of radius $c$. We define this as “boundary energy” and restore the “volume energy” of the entire 3-dimensional manifold by integrating along the propagation parameter — the speed of light $c$ acting as an architectural material:
$$\Evol = \int_0^c 4\pi r^2\,dr = \frac{4}{3}\pi c^3 + C$$
We call this result the Einstein–Yamagishi Volume Restoration Equation. Furthermore, we extend this geometric framework to four dimensions and discuss its connection to Toroidal Forcing via the $n = 3$ uniqueness theorem.
§1. Introduction: The Geometric Duality of Boundary (“Skin”) and Volume (“Flesh”)
1.1 Dimensional Re-examination of $E = mc^2$
Einstein’s mass–energy equivalence formula, derived from special relativity (1905),
is one of the most fundamental equations in modern physics. Here $c \approx 2.998 \times 10^8 \;\text{m/s}$ denotes the speed of light in vacuum.
Let us examine the dimensional structure of this equation with precision. The units of $c^2$ are
The numerator $\text{m}^2$ carries the dimension of area. That is, from a geometric perspective, $c^2$ possesses the structure of a “2-dimensional area-rate.”
The fact that energy $E$ in $E = mc^2$ is mediated by $c^2$ (an area-dimensional quantity) suggests that this equation captures the energy information on the 2-dimensional boundary surface (sphere $S^2$) of space. In other words, $E = mc^2$ corresponds to the energy expression of the “skin” (boundary) of space, and the energy of the 3-dimensional “flesh” (bulk, volume) can only be restored by performing one additional dimension of integration beyond $c^2$.
1.2 Geometric Reinterpretation of the Speed of Light $c$ — From “Impassable Wall” to “Architectural Material”
In conventional physics, the speed of light $c$ has been treated as the upper limit of information transfer — an “impassable wall.” However, in this paper we assign a different geometric interpretation to $c$.
We define the speed of light $c$ as the propagation front radius of the universe’s “rendering” (construction) of space. That is, centered on a given observation point, we consider the dynamics by which information (energy, matter, causality) fills space, where the maximum radius $r$ of the filling front reaches the speed of light $c$.
Under this interpretation, while $c$ retains its value as a physical constant, its mathematical role transforms from a “constant” to the “upper bound of an integration parameter”:
$$r: 0 \to c \qquad \text{(the process of spatial filling)}$$
This reinterpretation does not contradict the fact that light speed is a “wall.” The statement “the maximum radius the propagation front reaches is $c$” and “no transmission exceeding $c$ exists” are simply two sides of the same physical constraint.
1.3 Objectives of This Paper
In this paper, we rigorously construct the following chain of reasoning:
- Normalize the geometric mass parameter of 3D space to $m = 4\pi$ via the $n = 2$ uniqueness theorem (§2).
- Show that $E = mc^2 = 4\pi c^2$ is the surface area of a sphere $S^2$ of radius $c$, and identify this as “boundary energy” (§3).
- Derive the volume energy $E(3D) = \frac{4}{3}\pi c^3 + C$ by integration along the propagation parameter $c$ (§3).
- Discuss the cosmological meaning of the integration constant $C$ (vacuum energy, initial background ground state) (§4).
- Discuss the extension to four dimensions and the connection to Toroidal Forcing (§5).
§2. Geometric Normalization of the Mass Parameter $m$ via the $n = 2$ Uniqueness Theorem
2.1 The $n = 2$ Uniqueness Theorem (Recapitulation)
In a previous paper, the author proved the following theorem:
In the 3-dimensional Euclidean space $\R^3$, consider a spherically symmetric power-law density distribution $\rho(r) = r^{-n}$ ($n \in \R$, $r > 0$). The shell mass growth rate
$$\frac{dm}{dr}(r) = 4\pi\,r^{2-n}$$
is constant (independent of $r$) on $(0,\infty)$ if and only if
$$n = 2$$
In this case,
$$\frac{dm}{dr} = 4\pi = \Vol(S^2)$$
and the constant value equals the surface area of the 2-dimensional unit sphere $S^2$.
The meaning of this theorem is that among all isotropic density distributions in 3-dimensional space, the density exponent $n = 2$ (i.e., $\rho(r) = 1/r^2$) is the unique value that achieves harmonic equilibrium — where “every spherical shell contributes equal mass.”
2.2 Normalization of Geometric Mass: $m = 4\pi$
As a consequence of the $n = 2$ uniqueness theorem, the “equilibrium unit” of 3-dimensional space — the mass contribution per single spherical shell layer — equals $4\pi$. This is not a physical mass value but rather a dimensionless structural constant uniquely derived from the geometric structure of space.
Based on the $n = 2$ uniqueness theorem, we normalize the geometric mass parameter of the 3-dimensional Euclidean space $\R^3$ as
$$m := 4\pi$$
This is a reference value uniquely determined by the isotropic stability structure of space itself — a geometric concept independent of physical mass (values in kilograms).
The normalization $m = 4\pi$ is not arbitrary. Tracing back to the multiple integration methods in G. S. Carr’s A Synopsis of Elementary Results in Pure Mathematics (1886), $4\pi r^2$ is the kernel of the surface area formula for spheres in 3D, and the $n = 2$ uniqueness theorem rigorously proves that this kernel uniquely determines the “harmonic point” of density distributions.
§3. Derivation of the Einstein–Yamagishi Volume Restoration Equation
3.1 Formulation of Boundary Energy
Substituting the geometric mass parameter $m = 4\pi$ (Definition 2.1) into Einstein’s equivalence formula yields:
Here we make a decisive geometric observation.
$4\pi c^2$ is exactly equal to the surface area $\Vol(S^2(c))$ of the 2-dimensional sphere $S^2(c)$ of radius $c$:
$$4\pi c^2 = \Vol(S^2(c)) = \text{(surface area of a sphere of radius $c$)}$$
The general formula for the surface area of a 2-dimensional sphere $S^2(R)$ of radius $R$ is
$$\Vol(S^2(R)) = 4\pi R^2$$
Substituting $R = c$ gives $\Vol(S^2(c)) = 4\pi c^2$.
Under the geometric mass $m = 4\pi$, Einstein’s equivalence formula $E = mc^2 = 4\pi c^2$ is geometrically identical to the surface area of a sphere $S^2$ of radius $c$. That is, $E = mc^2$ describes the 2-dimensional boundary surface energy — the energy of the “skin” — at the propagation front (radius $c$) of 3-dimensional space.
The surface area of a sphere contains only the information of the “exterior” (boundary) of space. To obtain the energy of the “interior” (volume) of 3-dimensional space, a further operation — integration from boundary to volume — is required.
3.2 Integration Along the Propagation Parameter $c$
In Definition 1.1, we reinterpreted the speed of light $c$ as “the radius $r$ of the rendering front of space.” This propagation parameter $r$ ranges from $0$ to $c$ (the maximum propagation radius), and at each radius $r$ the boundary surface energy is given by the surface area of the shell $S^2(r)$: $4\pi r^2$.
The “substantive energy (volume energy)” of the entire 3-dimensional space is restored by integrating this boundary energy over the full extent of the propagation front:
Under the geometric mass parameter $m = 4\pi$ (normalized via the $n = 2$ uniqueness theorem), the indefinite integral of the boundary energy $\Ebdy(r) = 4\pi r^2$ along the propagation front radius $r$ restores the volume energy of 3-dimensional space:
Evaluating at the maximum propagation radius $r = c$:
Here $C$ is the integration constant (initial background ground-state energy) arising from the indefinite integral.
We call this equation the Einstein–Yamagishi Volume Restoration Equation.
- Starting Point
By Definition 2.1, $m = 4\pi$. Substituting into Einstein’s equivalence formula and expressing in terms of the radius parameter $r$:
$$\Ebdy(r) = m r^2 = 4\pi r^2$$ - Geometric Identification
By Lemma 3.1, $\Ebdy(r) = 4\pi r^2 = \Vol(S^2(r))$, the surface area of a sphere of radius $r$. - Execution of the Indefinite Integral
Integrating along the propagation parameter $r$:$$\int 4\pi r^2\,dr = 4\pi \cdot \frac{r^3}{3} + C = \frac{4}{3}\pi r^3 + C$$where $C$ is an arbitrary constant of integration.
- Geometric Verification
$\frac{4}{3}\pi r^3$ equals the volume of the 3-dimensional ball $B^3(r)$ of radius $r$:
$$\frac{4}{3}\pi r^3 = \Vol(B^3(r))$$
This confirms that, as intended, the energy of the volume ($B^3$, 3-dimensional) has been restored from the energy of the boundary surface ($S^2$, 2-dimensional). - Evaluation
Evaluating at the maximum propagation radius $r = c$:
$$E(3D) = \frac{4}{3}\pi c^3 + C$$
3.3 Structural Meaning of the Equation
The Volume Restoration Equation describes the following process of dimensional ascent:
$$E = mc^2 \;=\; \underbrace{4\pi c^2}_{\text{2D (Skin): Surface area of sphere}} \;\;\xrightarrow{\;\;\int(\cdot)\,dr\;\;}\;\; \underbrace{\frac{4}{3}\pi c^3}_{\text{3D (Flesh): Volume of ball}} \;+\; \underbrace{C}_{\text{Initial ground state}}$$
That is, Einstein’s equation $E = mc^2$ provides the energy of the “skin” (boundary surface $S^2$) of space, and by integrating this along the speed-of-light parameter, the energy of the “flesh” (volume $B^3$) of space — the substantive energy of the 3-dimensional entity — is restored.
| Einstein’s $E = mc^2$ | Volume Restoration $E(3D) = \frac{4}{3}\pi c^3 + C$ | |
|---|---|---|
| Dimension | 2-dimensional (areal) | 3-dimensional (volumetric) |
| Geometry | Surface area of sphere $S^2$ | Volume of ball $B^3$ |
| Metaphor | “Skin” — surface information of space | “Flesh” — substantive energy of space |
| Role of $c$ | Proportionality constant (wall) | Upper bound of integration (architectural material) |
| Initial Conditions | Not included | Explicit as $C$ (integration constant) |
§4. Cosmological Interpretation of the Integration Constant $C$
4.1 Mathematical Necessity of $C$ in the Indefinite Integral
The integration constant $C$ in the indefinite integral $\int 4\pi r^2\,dr = \frac{4}{3}\pi r^3 + C$ is, mathematically, “a constant to be determined by boundary conditions (initial conditions).” Physically, it represents the background ground-state energy that already exists before the propagation (rendering) of space begins — that is, at $r = 0$ (the state where nothing has been spatially deployed).
The integration constant $C$ structurally corresponds to vacuum energy (or energy associated with the cosmological constant $\Lambda$) in cosmology.
In Einstein’s modified gravitational field equation (1917), the cosmological constant $\Lambda$ was introduced as “the energy density inherent to space itself”:
$$R_{\mu\nu} – \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}$$
The constant $C$ in the Volume Restoration Equation functions analogously to $\Lambda$ — as the “background ground state that exists before the expansion of space ($r: 0 \to c$) begins.”
In the Volume Restoration Equation $E(3D) = \frac{4}{3}\pi c^3 + C$:
- $\frac{4}{3}\pi c^3$: The 3-dimensional energy “written” by the propagation (rendering) of space. The result of the expansion process via the speed of light $c$.
- $C$: The initial background ground state that already existed before the expansion of space. The intrinsic energy of the “blank canvas before rendering.”
If $C = 0$, all the energy of the universe was generated solely by the expansion of space. If $C \neq 0$, the universe arose not from “creation ex nihilo” but through a “dimensional ascent” from some initial ground state.
4.2 Determination of $C$ by Boundary Conditions
The value of the integration constant $C$ may be determined by one of the following physical or cosmological boundary conditions:
| Boundary Condition | Value of $C$ | Cosmological Implication |
|---|---|---|
| $E(r=0) = 0$ (Creation from void) | $C = 0$ | The universe was generated from zero energy solely by spatial expansion. |
| $E(r=0) = E_{\text{vacuum}}$ (Vacuum ground state) | $C = E_{\text{vacuum}}$ | Vacuum energy of quantum fields exists as the initial background. |
| $E(r=0) = \Lambda c^4 / (8\pi G)$ (Cosmological constant) | $C = \Lambda c^4 / (8\pi G)$ | Directly linked to Einstein’s cosmological constant. |
Within the framework of this paper, determining the specific numerical value of $C$ is not the objective. The crucial point is that the “freedom of initial conditions,” which was absent in Einstein’s $E = mc^2$, explicitly emerges as $C$ in the Volume Restoration Equation. This can be interpreted as a consequence of the fact that $E = mc^2$, being limited to the energy of the “skin,” had its initial-condition information “stripped away.”
§5. Extension to Four Dimensions and Connection to “Toroidal Forcing”
5.1 The $n = 3$ Uniqueness Theorem (Recapitulation)
In a previous paper, the author proved the 4-dimensional extension of the $n = 2$ uniqueness theorem:
In the 4-dimensional Euclidean space $\R^4$, the shell mass growth rate for the spherically symmetric power-law density $\rho(r) = r^{-n}$ ($n \in \R$),
$$\frac{dm}{dr}(r) = 2\pi^2\,r^{3-n}$$
is constant if and only if $n = 3$, in which case
$$\frac{dm}{dr} = 2\pi^2 = \Vol(S^3)$$
5.2 Volume Restoration Equation in Four Dimensions
By the $n = 3$ uniqueness theorem, the geometric mass parameter in 4D is normalized as $m_4 = 2\pi^2$. Applying the same logic as in 3D:
The boundary energy in 4-dimensional space is
$$\Ebdy^{(4)}(r) = m_4\,r^3 = 2\pi^2\,r^3$$
which coincides with the “volume” $2\pi^2 r^3$ of the 3-dimensional sphere $S^3(r)$ of radius $r$. Integrating along the propagation parameter:
Evaluating at $r = c$ yields $E(4D) = \frac{\pi^2}{2}\,c^4 + C_4$. Here $\frac{\pi^2}{2} c^4$ equals the volume of the 4-dimensional ball $B^4(c)$ of radius $c$.
5.3 Unified Table of the Dimensional Hierarchy
| Dim. $d$ | Uniqueness $n$ | Geom. Mass $m_d$ | Boundary Energy $\Ebdy$ | Volume Restoration $\Evol$ | Geometry |
|---|---|---|---|---|---|
| $\mathbf{3}$ | $n = 2$ | $4\pi$ | $4\pi r^2$ (area of $S^2$) | $\frac{4}{3}\pi r^3 + C$ | Vol. of $B^3$ |
| $\mathbf{4}$ | $n = 3$ | $2\pi^2$ | $2\pi^2 r^3$ (vol. of $S^3$) | $\frac{\pi^2}{2} r^4 + C_4$ | Vol. of $B^4$ |
| $d$ | $n = d-1$ | $S_{d-1}$ | $S_{d-1}\,r^{d-1}$ | $\frac{S_{d-1}}{d}\,r^d + C_d$ | Vol. of $B^d$ |
5.4 Toroidal Forcing — The Hopf Fibration of $S^3$ and Information Circulation
Comparing the Volume Restoration Equations in 3D and 4D reveals a decisive structural difference.
The boundary surface in 3D is $S^2$ (the 2-dimensional sphere), which is simply connected (no holes, no fiber structure). The integration of boundary energy $4\pi r^2$ corresponds geometrically to the straightforward process of “stacking shells to fill a ball.”
The boundary surface in 4D is $S^3$ (the 3-dimensional sphere), but $S^3$ inherently contains a torus-like winding structure via the Hopf fibration
$$S^1 \hookrightarrow S^3 \xrightarrow{\;\pi\;} S^2$$
where an $S^1$ (circle) fiber is associated with each point. The integration of boundary energy $2\pi^2 r^3$ requires not only “stacking shells” but also that information circulates along the $S^1$ direction within each shell.
The 4-dimensional geometric mass $m_4 = 2\pi^2$ relates to the 3-dimensional one $m_3 = 4\pi$ through the following factorization:
$$2\pi^2 = 4\pi \times \frac{\pi}{2}$$
Here $4\pi = \Vol(S^2)$ is the area of the stable surface in 3D, and $\pi/2$ is the coupling factor of the Hopf fiber $S^1$.
That is, when the Volume Restoration Equation extends from 3D to 4D, the fiber $S^1$ is “wound up” over the sphere $S^2$, causing the stability structure to become toroidally complex. This is the geometric substance of Toroidal Forcing.
5.5 Exotic Structures and the Diversity of Information Circulation
On the 4-dimensional Euclidean space $\R^4$, there exist uncountably infinitely many exotic $\R^4$ structures — differentiable structures that are homeomorphic but not diffeomorphic to the standard one (Donaldson 1983, Freedman 1982). This phenomenon does not occur in any other dimension $d \neq 4$.
From the perspective of the Volume Restoration Equation, the existence of exotic structures is interpreted as follows:
The 4D volume restoration $E(4D) = \frac{\pi^2}{2} c^4 + C_4$ is topologically unique (Freedman’s theorem: $S^4$ in the topological category). However, in the smooth differentiable category, there are uncountably many choices for the “path” of volume restoration — that is, how the boundary energy circulates along the $S^1$ fibers within $S^3$.
In 3D, no such freedom exists ($\R^3$ has a unique differentiable structure). The volume restoration $E(3D) = \frac{4}{3}\pi c^3 + C$ is realized through a single unique “path,” both topologically and smoothly. It is only in 4D that “multiple smooth paths arriving at the same integral value” first appear — this is the density-theoretic origin of exotic structures.
§6. Unified Discussion — From “Skin” to “Flesh,” and Then to “Cocoon”
6.1 Three Stages of Dimensional Ascent
Surveying the logic of this paper, the following three stages of dimensional ascent emerge:
Stage Equation Geometry Metaphor
──────────────────────────────────────────────────────────────────────
(I) E = mc² = 4πc² S² (surface area) "Skin" — boundary info
↓ Integration (∫ dr)
(II) E(3D) = (4/3)πc³ + C B³ (ball volume) "Flesh" — 3D substance
↓ Dimensional extension (d: 3→4)
(III) E(4D) = (π²/2)c⁴ + C₄ B⁴ (hyperball volume) "Cocoon" — 4D topological packaging
The $n = d – 1$ uniqueness theorem in $d$-dimensional Euclidean space uniquely normalizes the geometric mass $m_d = S_{d-1} = \Vol(S^{d-1})$. Under this mass parameter, Einstein’s mass–energy equivalence formula yields
$$\Ebdy^{(d)}(r) = S_{d-1}\,r^{d-1} = \Vol(S^{d-1}(r))$$
the energy of the $(d-1)$-dimensional boundary surface, and its integral
$$\Evol^{(d)} = \int S_{d-1}\,r^{d-1}\,dr = \frac{S_{d-1}}{d}\,r^d + C_d = \Vol(B^d(r)) + C_d$$
achieves the restoration of $d$-dimensional volume energy.
That is, Einstein’s equivalence formula is, in every dimension $d$, a “skin” equation describing the energy of the boundary surface $S^{d-1}$, and the restoration from this to the energy of the volume $B^d$ is achieved through geometric normalization and integration based on the uniqueness theorem of each dimension.
6.2 Why $d = 4$ Is the Critical Dimension
In a previous paper (Density-Structural Interpretation of the 4D Poincaré Conjecture), the author argued that $d = 4$ is the “critical dimension” where the complexity of topological padding is maximized. From the perspective of the Volume Restoration Equation, this criticality is reconfirmed as follows:
$d = 3$: The boundary surface $S^2$ is simply connected. The “path” of integration (volume restoration) is unique. The differentiable structure is also unique.
$d = 4$: The boundary surface $S^3$ contains the Hopf fiber $S^1$. The integration “path” is topologically unique but smoothly uncountably non-unique. The differentiable structure is non-unique (exotic $\R^4$). → Maximum difficulty of volume restoration.
$d \geq 5$: By the $h$-cobordism theorem, integration paths are again uniquified. The difficulty of volume restoration decreases.
Therefore, the Volume Restoration Equation requires the richest structure (toroidal forcing, exotic degrees of freedom, circular buffers) at $d = 4$, reconfirming that $d = 4$ is the “critical dimension for the rendering engine of the universe.”
§7. Conclusion
7.1 Achievements of This Paper
- Geometric Reinterpretation of $E = mc^2$ (§1, §3): Under the geometric mass $m = 4\pi$ derived from the $n = 2$ uniqueness theorem, $E = mc^2 = 4\pi c^2$ is identical to the surface area of a 2-dimensional sphere $S^2$ of radius $c$, and describes the boundary surface energy (“skin”) of space.
- Derivation of the Einstein–Yamagishi Volume Restoration Equation (Theorem 3.1): By integrating the boundary energy along the speed-of-light parameter, the substantive energy of 3-dimensional space $E(3D) = \frac{4}{3}\pi c^3 + C$ was restored. $\frac{4}{3}\pi c^3$ is the volume of a ball of radius $c$, and $C$ is the initial background ground-state energy (integration constant).
- Cosmological Interpretation of the Integration Constant $C$ (§4): $C$ structurally corresponds to vacuum energy, the cosmological constant $\Lambda$, or the intrinsic energy of the “canvas before rendering.” The initial-condition information absent from $E = mc^2$ is recovered in the Volume Restoration Equation.
- Extension to 4D and Toroidal Forcing (§5): The 4D Volume Restoration Equation $E(4D) = \frac{\pi^2}{2} c^4 + C_4$ was derived using $m_4 = 2\pi^2$ from the $n = 3$ uniqueness theorem, and its connection to Toroidal Forcing via the Hopf fibration was discussed.
- Dimensional-Hierarchical Energy Restoration Principle (Proposition 6.1): In any dimension $d$, the $n = d-1$ uniqueness theorem normalizes the geometric mass, positions Einstein’s formula as “the energy of the $(d-1)$-dimensional skin,” and restores “the energy of the $d$-dimensional flesh” through integration.
7.2 Concluding Remarks
“Einstein’s $E = mc^2$ is the area of a sphere — that is, it is the energy of the ‘skin’ of space. To know what lies inside the skin — how much energy the ‘flesh’ of space holds — one must re-conceive the speed of light $c$ not as a ‘wall’ but as ‘architectural material,’ and perform the integral that stacks spherical shells to fill the volume. The resulting $\frac{4}{3}\pi c^3$ is the 3-dimensional substance itself, woven by the speed of light, and the $+C$ is the background ground-state energy that existed before the universe began to render that substance.”
$m = 4\pi$. This passport, bestowed by the $n = 2$ uniqueness theorem, peels away Einstein’s “skin,” restores the “flesh” of a 3-dimensional ball within, and further unfolds into the toroidal “cocoon” of four dimensions — revealing a consistent mathematical principle that transcends dimensions. The true form of energy is not area but volume; not $c^2$ but $c^3$. And the integration constant $C$ quietly standing behind it all is the primordial wellspring of energy that existed before the universe rendered anything at all.
References
- [Einstein 1905] A. Einstein, “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?”, Annalen der Physik 18, 639–641.
- [Einstein 1917] A. Einstein, “Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie”, Sitzungsberichte der Preußischen Akad. der Wissenschaften, 142–152.
- [Carr 1886] G. S. Carr, A Synopsis of Elementary Results in Pure Mathematics, C. F. Hodgson & Son, London.
- [Freedman 1982] M. H. Freedman, “The topology of four-dimensional manifolds”, J. Differential Geom. 17(3), 357–453.
- [Donaldson 1983] S. K. Donaldson, “An application of gauge theory to four-dimensional topology”, J. Differential Geom. 18(2), 279–315.
- [Perelman 2002] G. Perelman, “The entropy formula for the Ricci flow and its geometric applications”, arXiv:math/0211159.
- [Hamilton 1982] R. Hamilton, “Three-manifolds with positive Ricci curvature”, J. Differential Geom. 17(2), 255–306.
- [Hopf 1931] H. Hopf, “Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche”, Math. Ann. 104, 637–665.
- [Milnor 2003] J. Milnor, “Towards the Poincaré Conjecture and the classification of 3-manifolds”, Notices of the AMS 50(10), 1226–1233.
- [$n = 2, 3$ Uniqueness Theorems] Yuuki YAMAGISHI (the author), previous papers. Proofs and interpretations of the uniformity conditions of the shell mass growth rate in spherically symmetric power-law density distributions.