The Chandrasekhar limit for white dwarfs is 1.44 Solar masses, however the heaveist known white dwarf is only 1.35 solar masses. https://earthsky.org/space/smallest-most-massive-white-dwarf/
What's the cause of this gap in mass?
Astrophysics – Why Are There No Known White Dwarfs Between 1.35 to 1.44 Solar Masses?
astrophysicsneutron-starswhite-dwarfs
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We think that most neutron stars are produced in the cores of massive stars and result from the collapse of a core that is already at a mass of $\sim 1.1-1.2 M_{\odot}$ and so as a result there is a minimum observed mass for neutron stars of about $1.2M_{\odot}$ (see for example Ozel et al. 2012). Update - the smallest, precisely measured mass for a neutron star is now $1.174 \pm 0.004 M_{\odot}$ - Martinez et al. (2015).
The same paper also shows that there appears to be a gap between the maximum masses of neutron stars and the minimum mass of black holes.
You are correct that current thinking is that the lower limit on observed neutron star and black hole masses is as a result of the formation process rather than any physical limit (e.g. Belczynski et al. 2012 [thanks Kyle]).
Theoretically a stable neutron star could exist with a much lower mass, if one could work out a way of forming it (perhaps in a close binary neutron star where one component loses mass to the other prior to a merger?). If one just assumes that you could somehow evolve material at a gradually increasing density in some quasi-static way so that it reaches a nuclear statistical equilibrium at each point, then one can use the equation of state of such material to find the range of densities where $\partial M/\partial \rho$ is positive. This is a necessary (though not entirely sufficient) condition for stability and would be complicated by rotation, so let's ignore that.
The zero-temperature "Harrison-Wheeler" equation of state (ideal electron/neutron degeneracy pressure, plus nuclear statistical equilibrium) gives a minimum stable mass of 0.19$M_{\odot}$, a minimum central density of $2.5\times10^{16}$ kg/m$^3$ and a radius of 250 km. (Colpi et al. 1993). However, the same paper shows that this is dependent on the details of the adopted equation of state. The Baym-Pethick-Sutherland EOS gives them a minimum mass of 0.09$M_{\odot}$ and central density of $1.5\times10^{17}$ kg/m$^3$. Both of these calculations ignore General Relativity.
More modern calculations (incorporating GR, e.g. Bordbar & Hayti 2006) get a minimum mass of 0.1$M_{\odot}$ and claim this is insensitive to the particular EOS. This is supported by Potekhin et al. (2013), who find $0.087 < M_{\rm min}/M_{\odot} < 0.093$ for EOSs with a range of "hardness". On the other hand Belvedere et al. (2014) find $M_{\rm min}=0.18M_{\odot}$ with an even harder EOS.
A paper by Burgio & Schulze (2010) shows that the corresponding minimum mass for hot material with trapped neutrinos in the centre of a supernova is more like 1$M_{\odot}$. So this is the key point - although low mass neutron stars could exist, it is impossible to produce them in the cores of supernovae.
Edit: I thought I'd add a brief qualitative reason why lower mass neutron stars can't exist. The root cause is that for a star supported by a polytropic equation of state $P \propto \rho^{\alpha}$, it is well known that the binding energy is only negative, $\partial M/\partial \rho>0$ and the star stable, if $\alpha>4/3$. This is modified a bit for GR - very roughly $\alpha > 4/3 + 2.25GM/Rc^2$. At densities of $\sim 10^{17}$ kg/m$^3$ the star can be supported by non-relativistic neutron degeneracy pressure with $\alpha \sim 5/3$. Lower mass neutron stars will have larger radii ($R \propto M^{-1/3}$), but if densities drop too low, then it is energetically favorable for protons and neutrons to combine into neutron-rich nuclei; removing free neutrons, reducing $\alpha$ and producing relativistic free electrons through beta-decay. Eventually the equation of state becomes dominated by the free electrons with $\alpha=4/3$, further softened by inverse beta-decay, and stability becomes impossible.
A back of the envelope calculation (and that is all this is) would go along the lines of assuming that the white dwarf is made entirely of $^{12}$C (it isn't) and is entirely converted into $^{56}$Ni (it isn't).
The appropriate mass to use would be $\sim 1.4M_{\odot}$ (it is actually a touch lower - the real "Chandrasekhar mass" at which instability sets in is determined by GR collapse; or by inverse beta decay; or by the onset of pyconuclear reactions, all of which take place at $\rho \sim 3 \times 10^{13}$ kg/m$^3$ when the white dwarf has a mass of about 1.37-1.38$M_{\odot}$).
If the star is entirely $^{12}$C, then this means $1.40 \times 10^{56}$ carbon nuclei, containing $1.68\times 10^{57}$ baryons. To conserve the baryon number, the number of $^{56}$Ni nuclei produced is smaller by a factor of 12/56.
The mass of each carbon nucleus (by definition) is $12m_u$, where $m_u$ is the atomic mass unit. The mass of each nickel nucleus is $55.94m_u$.
Thus the change in mass converting all the carbon into nickel is $$ \Delta M \simeq 1.40\times10^{56}\times 12m_u - 1.40\times10^{56}\times (12/56)*55.94m_u$$ $$\Delta M \simeq 1.8\times 10^{54} m_u = 3.0\times10^{27}\ {\rm kg}$$
Converting this to energy gives $2.7\times 10^{44}$J, which is indeed roughly the energy involved in a type Ia supernova. This is what is responsible for "exploding" the star, since with an initial radius of $\sim 1000$ km, it has a gravitational binding energy, $\sim -3GM^2/5R = -3\times 10^{44}$ J.
A slightly less back of the envelope calculation would include the internal energy of the relativistic electrons, which shrinks the magnitude of the binding energy considerably (it would be exactly zero for a star entirely governed by ideal ultra-relativistic degeneracy pressure and halved for non-relativistic degeneracy pressure), so that a large fraction of the energy released can actually go into photons, neutrinos and the kinetic energy of the ejecta.
Best Answer
Two reasons. Firstly, the "Chandrasekhar mass" of 1.44 solar masses is based on a pair of unrealistic assumptions, that are not met in practice, which means the true mass limit is more like 1.37 or 1.38 solar masses. Secondly, white dwarfs more massive than about $1.2 M_{\odot}$ are not produced by normal single-star stellar evolution, only through mass transfer in binary systems. This mass transfer may result in the star exploding as a supernova before it grows beyond $1.35M_{\odot}$.
The two assumptions are: (I) that the white dwarf is supported by ideal electron degeneracy pressure. i.e. Point-like, non-interacting fermions. (II) That the structure of the star is governed by Newtonian gravity.
The first assumption fails because the electrons and ions do have Coulomb interactions that make the material more compressible. More importantly, at high densities (and the density increases with mass), the electron Fermi energy eventually becomes high enough to initiate electron capture to make more neutron-rich nuclei. Since the electrons are ultra-relativistic, the star is already marginally stable at this stage, and the removal of electrons causes instability and collapse.
The second assumption fails because more massive white dwarfs are smaller and General Relativity must be used. The General Relativistic formulation of the equation of hydrostatic equilibrium features pressure on the RHS. So the higher the pressure, the steeper the required pressure gradient. Ultimately, this also leads to an instability at a finite size and density that occurs at masses lower than the canonical Chandrasekhar mass.
For typical C/O white dwarfs, both of the instabilities discussed above occur when the white dwarf is at about 1.38 solar masses.
Note that white dwarfs of more than about 1.2 solar masses are not expected to arise from the evolution of a single star. If the C/O core of a star is more massive than this, then it will also become hot enough to ignite these elements. More massive white dwarfs will need to have been produced by accretion in a binary system or by a merger. Then, another factor comes into play, which is the possible detonation of the entire white dwarf, which may also occur above 1.35 solar masses, possibly ignited by the fusion of helium from the accreted material or by pycnonuclear reactions as the C/O core increases in density.
Postscript - there actually are some white dwarfs with estimated masses of $1.35-1.37M_\odot$ in classical novae binary systems (e.g. Hachisu & Kato 2001). These may be systems that are about to go "bang".